this video is going to be for those of you who are studying for your final exam in algebra so let's go over this problem which of the following answer choices is equivalent to the expression shown below now for each of these problems i recommend that you pause the video and work on it yourself and then when you're done you can play the video again to see if you have the right answer so let's start with the first one we need to multiply two binomials together so we need to use the foil method so f stands for first we're going to multiply 3x by 4x 3x times 4x is 12x squared and then we're going to multiply this stuff on the outside 3x times negative 7 that's negative 21x and then the stuff on the inside 5 times 4x is positive 20x and finally the last ones 5 times negative 7 is negative 35. now let's go ahead and combine like terms negative 21 plus 20 is negative 1x which we can simply write as negative x so our final answer is 12x squared minus x minus 35. so this corresponds to answer choice b number two which of the following is equivalent to the difference of the two polynomials shown below is it a b c or d well the first thing we need to do is get rid of the parentheses and distribute the negative sign since we don't have anything in front of the first set of parentheses we can just get rid of it so this is equal to nine x cubed minus five x squared plus seven now we do have a negative sign in front of the second set of parentheses so we do need to distribute it so we're going to have negative 4x cubed and then plus 8x and then a negative times a negative that's a positive so this is going to be plus nine now let's combine like terms so these two they're similar they both contain an x cube nine minus four is five now these two terms are not like terms this carries an x squared this one doesn't so we can't add them all we can do is just write them the way they are now these two are like terms so we can add seven and nine which will give us sixteen and so this is it it's five x cubed minus five x squared plus eight x plus sixteen and so that corresponds to answer choice c now what about this one what is the slope of the line that passes through the points 3 comma negative 5 and negative 9 comma negative one the formula that we need for this problem is this equation the slope is y two minus y one divided by x two minus x one so for the first point x1 is going to be 3 y1 is negative 5. x2 and y2 represents the coordinates for the second point so y2 is negative 1 and then minus y one which is negative five x two is negative nine and then minus x one which is positive three so what is negative one minus negative five whenever you have two negative signs next to each other they will change it to a positive sign so negative one plus five that's going to be four and then negative nine minus three is going to be negative twelve if you have a negative and a positive it becomes a negative sign so this is just negative nine minus three our next step is to reduce the fraction negative 12 is basically negative four times three and so we could cancel a four four divided by four is one and we could put the negative on the top of the fraction so this simplifies to negative one-third which means that answer choice e is the right answer number four which of the following answer choices is equal to the expression shown below so for this one we need to use the order of operations particularly perhaps you heard of this term pemdas please excuse my dear unsally so first we need to work on the stuff inside the parentheses two minus nine is negative seven next we need to work on the exponents negative seven squared basically means that we're multiplying two negative sevens together next we can mult we can work on multiplication negative seven times negative seven is positive forty nine and three times forty nine well three times fifty is one fifty and if you take away a three from that that will give you one forty seven now the last thing we need to do is add 8 plus 147 is 155. so this is the answer which corresponds to answer choice d number five factor each expression completely so this is a free response problem and let's start with the first one how can we factor 9x squared minus 64. now this is a formula that can help you and it's the difference of perfect squares a squared minus b squared is going to be equal to so what you do is you take the square root of a squared which is a and then take the square root of b squared which is b and then one of them will be a plus the other will be a minus so let's apply the same technique to this problem so what is the square root of nine x squared the square root of nine is three the square root of x squared is x so in this case a is 3x now if b squared is 64 what is b what is the square root of 64. the square root of 64 is eight so the answer is going to be three x plus eight and three x minus eight so that's how you can factor an expression using difference of using the difference of squares method now let's move on to the second one so here we have a trinomial with a leading coefficient of one how can we factor this expression for a situation like this here's a quick and simple method find two numbers that multiply to 12 but that add to the middle coefficient of seven so what two numbers multiply to twelve but add to seven we can list out the factors of twelve one times twelve is twelve two times six is twelve and three times four is twelve only three and four will add up to seven so this is what we need three plus four is seven three times 4 is 12. so all we got to do is write x plus 3 times x plus 4. and so that's how we can factor this expression to check your work what you could do is foil this expression x times x is x squared x times four is four x three times x is three x three times four is twelve and so this is going to give us x squared plus 7x plus 12. so this right here is the answer that's how you can factor this trinomial now let's move on to the third example six x squared plus x let's write that as one x minus fifteen so this time we have a trinomial where the leading coefficient is not one it's six so the method that we're going to use to factor this expression is going to be different than the one we use for number two so what's our first step the first thing you want to do is multiply the leading coefficient by the constant term so you want to multiply 6 and negative 15. six times negative fifteen is six times ten is sixty six times five is thirty sixty plus thirty is ninety so this will give us negative ninety now what two numbers multiply to negative 90 but add to positive one if you're not sure what you could do is make a list negative 90 divided by one is negative 90. if you take a negative 90 divided by 2 you get negative 45. if you divide it by 3 you get negative 30. now negative 90 divided by 4 it doesn't give us a nice number so we're going to skip that if we divided by 5 it's negative 18. if we divided by 6 negative 15 7 and 8 doesn't go into 90. if we divided by 9 we'll get negative 10 and if we divide it by 10 we get negative nine once the numbers begin to reverse you can stop at that point now which pair of numbers will add up to one so looking at the list if we look at nine and negative ten that's going to add up to negative one but positive ten and negative nine adds up to positive one and so this is what we're going to use now here's how we're going to use those numbers we're going to replace 1x with 10x and negative 9x so you can write it as 10x minus 9x or you could say 9x plus 10x it really doesn't matter so i'm going to write 10x and negative 9x our next step is to factor by grouping and the way you do this is you take out the gcf the greatest common factor in the first two terms the greatest common factor is going to be two x six x squared divided by two x is three x ten x divided by two x is five now let's take out the gcf and the last two terms this is going to be negative three negative nine x divided by negative three is three x negative fifteen divided by negative three is positive five now notice that these two expressions are the same if you see that that means you're on the right track so what we're going to do is factor out the gcf once again the gcf is these two expressions 3x plus 5. so we're going to write it once if you take this expression divided by 3x plus 5 the three x plus five terms will cancel and you're gonna you're gonna get two x if you take this divided by three x plus five you're gonna get the number on the outside which is negative three and so this here is the answer it's equivalent to our original expression and so that's how you can factor 6x squared plus x minus 15. it's going to be 3x plus 5 times 2x minus 3. and you could foil this expression to make sure that it's equal to this now the last one that we're going to talk about number four is 27.5 x cubed minus 64. so what can we do to factor this expression it looks like the first one but it's a little different notice that you could take the square root of 9 and 64. so this is a difference of squares notice that you could take the cube root of 27 and 64. but it's not easy to take the square root of 27 because it won't give us a nice whole number so this is the difference of perfect cubes situation and the formula that we need to apply in this case is a cube minus b cube and that's equal to a minus b times a squared plus a b plus b squared so what is a and what is b here's a question for you if a to the third is 27 x cubed what is a well a has to be the cube root of this the cube root of 27 is 3 and the cube root of x cubed is simply x so a is equal to 3x now what is b b to the third is equal to 64. not negative 64 but positive 64. this negative sign belongs here the cube root of 64 is 4 because 4 times 4 times 4 is 64. so we have b by the way for those of you who want more practice examples on factoring or if you need help with a topic that you don't see in this video go to youtube and in the search box type in new algebra playlist and then type in my channel name organic chemistry tutor my algebra playlist will come up and you can find more examples on factoring or other topics in algebra that you might need help with so feel free to take a look at that when you get a chance now let's go back to this one so we have the left side of the formula here all we need to do is complete the right side now that we know what a and b are equal to so a is three x and then minus b which is just four a squared that's a times a so three x times three x is nine x squared a b that's 3x times 4 that's going to be 12. and there's a plus sign between it it's going to be 12x rather and then plus b squared so 4 squared is 4 times four which is sixteen and so this is the answer now let's move on to number six which of the following is a solution to the equation shown below now whenever you wish to solve an equation you need to isolate the variable you need to get it on one side of the equation all the other numbers you want to move to the other side and once you get that variable by itself whatever is on the other side that's what x is going to equal to so let's begin the first thing we want to do is distribute we want to get rid of the parentheses so here we have 3 times x which is going to be 3x and then 3 times negative 15 i mean 3 times negative 5 is negative 15. and then we have 5 times x and then 5 times 3 which is 15. now our next step is to combine like terms negative 15 plus 7 that's going to be negative 8 and then 15 minus 41 is negative 26. now let's add 26 to both sides and simultaneously let's subtract both sides by 3x if we do this notice that the x variables on the left will cancel and so what we're going to have right now is 5x minus 3x which is 2x and negative 8 plus 26 which is 18. so now the x variable is only on one side of the equation now we need to get rid of the two and one way in which we can do so is by dividing both sides by two eighteen divided by two is nine two x divided by two is x and so we get the answer x is equal to nine which means answer choice c is the answer number seven which of the following equations corresponds to the graph shown below so looking at our answer choices we could see that all of them are in slope intercept form that is y equals mx plus b format the number in front of x is the slope and b is the y-intercept now what is the y-intercept for this particular line what is it basically it's where the line touches the y-axis so the vertical axis is the y-axis the horizontal one is the x-axis and we can see that every mark represents one so on the y-axis this is negative one negative two negative three negative four so the y-intercept our b value is negative four which means we can eliminate answer choice d because here the y-intercept is positive four now the next thing we need to consider is the slope here we have a slope of negative 2 negative 3 over 2 and positive 3 over 2. now what is the difference between a positive slope and a negative slope a line with a positive slope will be going up as you move to the right a line with a negative slope will be going down as you move to the right so this line would you say it has a positive slope or a negative slope as we move to the right notice that it goes up so the slope for this line is positive understanding that helps us to eliminate answer choice a and b because they both contain a negative slope they should have a line that goes down as you move to the right so therefore answer choice c has to be the correct answer now another way in which you can get the exact slope is by finding another point on this line so we have the y intercept which is that negative it's at zero negative four and we can see that there's another point here which is at a y value of negative one and an x value of two now recall that the slope is basically the rise divided by the run so to go from the first point which we'll call p1 to the second point p2 we need to go up three units from negative four to negative one so that's the rise we're increasing by positive three and the run we need to travel by two units to the right so the slope is three over two therefore you can clearly see that c is the right answer number eight simplify the expression shown below now before we do this problem it's best to go over some rules when dealing with exponents let's say if we're multiplying x squared by x cubed what should we do whenever you're multiplying two common bases you can add the exponents so two plus three is five another way to see this is that x squared basically means that you're multiplying two x variables together x cubed means that you're multiplying three x variables together and so you get x to the fifth because you're multiplying a total of five x variables together so that's the first rule we need to know when multiplying common bases in this case x you can add the exponents now what about dividing let's say if we're dividing x to the seven by x to the fourth when you're dividing you can subtract so you take the top exponent and subtract it by the bottom one so 7 minus 4 is 3. you get x cubed now to confirm this you can expand the expression x to the seven means that we're multiplying seven x variables together and x to the fourth means that we're multiplying four x variables together so we could cancel four x variables on the top and on the bottom which leaves us with three x variables on the top and so this becomes x cubed because we have three left over now what about if we raise one exponent to let's say another exponent so here we have x squared raised to the third power what should you do when you raise one exponent to another exponent you need to multiply so two times three is six this becomes x to the sixth power now another way to see this is to understand that this three tells us that we have three x squareds being multiplied to each other and each x squared is basically two x variables multiplied to each other so notice that we have a total of six x variables thus this whole thing is equal to x to the sixth so whenever you raise one exponent by another exponent you need to multiply so now let's work on this example the exponent on two and three is 1. 2 to the first power is just 2. so what we're going to do is distribute this exponent to all the exponents on the inside so here we're going to have 3 times 1 which is 3 so this is going to become two to the third power and then three times three is nine so x to the ninth power and three times four is twelve y to the twelfth power now let's do the same thing here four times one is four and then four times two is eight and finally four times five is twenty now what is two raised to the third power what is that equal to that's 2 times 2 times 2 so basically you're multiplying 2 3 times 2 times 2 times 2 is 8. now 3 to the fourth power three times three times three times three let's write that out now these two threes will give us nine and the same is true for those two threes and then nine times nine is 81 so 3 to the fourth power is 81 now let's multiply y i mean excuse me x to the ninth power by x to the eighth power when you're multiplying common bases you can add the exponents so nine plus eight is seventeen and let's also multiply eight by eighty one eight times 81 that's going to be 648 now for those of you who for some reason can't use a calculator you can go back to the old school way of doing math 8 times 1 is 8 and then 8 times 8 is 64. and so you get 648 now let's multiply y to the 12 by y to the 20. so we need to add 12 plus 20 and that's going to give us 32 so this right here is the final answer 648 x to the 17th power y to the 32 that's it number nine simplify the expression shown below so negative three times five x squared y to the seventh z to the four raised to the zero power what is that equal to now you need to know that anything raised to the zero power is one x to the zero is one five to the zero is one so anything raised to the zero power is one so what's the final answer for this problem the final answer is not one everything within the parentheses will become one because it's affected by the exponent which is zero so this entire portion highlighted in blue that is equal to one now we still have a negative three on the outside so it's going to be one times negative three so the final answer for this problem is negative three now let's try this one simplify the expression now what do you recommend should we simplify the stuff within the parentheses and then raise it to the second power or is it better to distribute the exponent and then simplify everything on the inside both ways can give you the right answer but in this particular case it's better to simplify before we apply the exponent 36 to the second power is going to be a big number and so i don't want to deal with that so let's simplify everything on the inside now i'm going to write 36 as 9 times 4 because 63 is also divisible by 9. i can write 63 as 9 times 7. now what is x to the fourth power divided by x to the 9th power so whenever you're dividing you need to subtract take the top number which is four and then subtract it by the bottom number which is nine four minus nine is negative five now you can put a one on the bottom if you want after you subtract it whatever result you get is initially on the top now when dealing with a negative exponent you need to move the variable to the bottom to make it positive so this is one over x to the fifth power now for those of you who wish to understand it you can do this so we can expand x to the fourth by writing four x variables and x to the ninth i'm gonna write out nine x variables so we can cancel four on top four on the bottom so we don't have any more x variables on top so we're just going to put a one and on the bottom we have five x variables so this is one over x to the fifth that's another way in which you can see it so let's put that on the bottom next we're going to divide y to the fifth by y to the negative three so take the top number which is five and then subtract it by the bottom number which is negative three five minus negative three is the same as five plus three and so this becomes eight because it's already positive well no matter what we do after you do the subtraction thing it's automatically on top but if you get a negative exponent then you move it to the bottom so we have a y to the eighth on the top now let's do the same thing with z so z to the negative four divided by z to the negative six so let's take the top number negative four and subtract it by the bottom number negative six so this is going to be negative 4 plus 6 which is positive 2. so because we have a positive exponent it's going to remain on the top now before we distribute the exponent let's cancel the nine nine divided by nine is one and now we could distribute two so i'm going to write this as four to the first power and seven to the first power two times one is two so we're gonna have four squared and then two times eight is sixteen and then two times two is four on the bottom we have two times one which is two and then five times two is ten so basically multiply all the exponents by two now four to the second power that's 4 times 4 which is 16 7 squared which is 7 times 7 that's 49 so this is the final answer 16 y raised to 16 times z raised to the 4 divided by 49 x to the 10th power number 11 which of the following is a solution to the equation shown below now whenever you have fractions separated by an equal sign what you want to do is you need to cross multiply so we're going to multiply 2 by x minus 1. and then we're going to multiply four by x minus three now let's go ahead and distribute so two times x minus one that's gonna be two x minus two and four times x minus three is four x minus twelve our next step is to subtract both sides by two x and at the same time we can add twelve to both sides so these will cancel negative 2 plus 12 is 10 4x minus 2x is 2x and so we have 2x is equal to 10 next we could divide both sides by 2. 10 divided by 2 is 5. and so that's it for this problem e is the right answer choice number 12 which of the following is a solution to the equation 6x squared minus 29x plus 28 equals zero so what techniques do you think we need to use in order to solve this equation now there's different things you can do if you don't know you can take one of the answer choices plug it in and see if you get zero that's one option another option is to factor we have a trinomial where the leading coefficient is not one it's six in this case so we can factor it or we can use the quadratic formula to get the answer so we're going to use the last two methods so let's factor it first let's multiply the leading coefficient by the constant term so what is six times twenty eight six times twenty is basically one twenty if you have six twenty dollar bills that's a hundred twenty dollars six times eight is forty eight so one twenty plus 48 that's going to be 168. now what two numbers multiply to 168 but add up to negative 29. so these are gonna have to be two negative numbers if you divide 168 by negative 1 you're going to get negative 168. if you divided by negative 2 you're going to get negative 84. and it's good to use a calculator at this point if you're allowed to do so now we could also divide it by three this will give us negative 56 and if we divide it by four it will give us a negative 42. so let's pick a bigger number let's try dividing it by 8. so that will give us 21. notice that these two numbers they add up to negative 29. so once you create a list you just have to find out which pair of numbers will add up to the middle coefficient now what we're going to do is we're going to replace negative 29x with negative 8x and negative 21x now let's factor by grouping so let's take out the greatest common factor in the first two terms in this case we could take out a 2x 6x squared divided by 2x is 3x negative 8x divided by 2x is negative 4. now in the next two terms let's take out the greatest common factor which is going to be negative seven negative 21x divided by negative seven that's going to give us positive 3x positive 28 divided by negative seven is negative four so next we need to factor out another gcf which is three x minus four so if we take out three x minus four from the first term we're going to have 2x left over and if we take it out in the second term we're going to have negative 7 left over now at this point we can set each factor equal to zero because if one of them is equal to zero the whole equation is true zero times anything is zero so we're gonna set 3x minus four equal to zero and two x minus seven equal to zero so let's focus on the first one the first thing we need to do is add four to both sides and so we're going to get 3x is equal to 4. next we need to divide both sides by 3. and so the first answer is what we have here x is equal to four over three now there's another possible solution so let's start with the second equation and let's begin by adding seven to both sides so we have 2x is equal to 7 and then we could divide both sides by 2. so x is equal to seven over two now the only answer that's listed is answer choice c x is equal to four over three so that's the correct answer now let's talk about how we can use the quadratic formula to get the same answer so i'm going to write the answers that we have on top so it was 4 over 3 and also 7 over 2. now here's the quadratic formula x is equal to negative b plus or minus the square root of b squared minus 4ac divided by 2a now you need to know that a is equal to 6 b is negative 29 c is 28 the equation is already in standard form that is it's an ax squared plus bx plus c form so now let's plug in those numbers into the formula so we said b is negative 29. and then we're going to have a b squared 29 squared minus 4 times a which is 6 times c which is 28 and then this is divided by 2a or 2 times 6. so this is going to change to positive 29 and 29 squared that is 840 one now four times six is twenty four twenty four times twenty eight is six hundred and seventy two and then we're going to divide that by twelve now eight 841-672 it's 169. the square root of 169 is 13. so we're going to have 29 plus or minus 13 divided by 12. so first let's do 29 plus 13 which is 42 and so the first answer is 42 over 12. the second one is going to be 29 minus 13 which is 16. so it could also be 16 over 12. so now we need to reduce these two fractions 42 over 12 can be written as 42 is basically 6 times 7 and 12 is 6 times 2. so if we cancel the six we get the answer x is equal to seven over two now for the other answer sixteen over we could say 16 is 4 times 4 12 is 4 times 3 so if we cancel the 4 we get the other answer 4 over 3. so if you don't want to factor you can also solve the quadratic equation using the quadratic formula once again c is the correct answer number 13 which of the following is a solution to the system of linear equations shown below so go ahead and take a minute and try that problem now you could use the substitution method or you could use the elimination method i'm going to use the elimination method i'm going to try to cancel the y variables right now i cannot add the two equations because they won't completely cancel now the common multiple of 2 and 3 is 6. so i need to get 6y in both equations i'm going to multiply the first equation by 3 because that's going to give me positive 6y and the second one by 2 because it's going to give me negative 6y which will add up to 0. so let's multiply everything in this equation by 3. 7x times 3 is 21x 2y times 3 is 6y and 24 times 3 is 72. now let's multiply everything in the second equation by 2. 5x times 2 is 10x negative 3y times 2 is negative 6y and negative 5 times 2 is negative 10. now we're going to add up the two equations so these will cancel 21x plus 10x is 31x and 72 plus negative 10 is 62. so now we can divide both sides by 31. and so this will give us the x value which is 2. because this is a multiple choice problem we can already see what the answer is we're going to write the answer as an x wide ordered pair so it has to be d because it's the only one with an x value of 2. but let's calculate y so once you have the x value you can plug it into the first or the second equation to get y let's use the first equation so let's replace x with two and then let's calculate y seven times two is fourteen next let's subtract both sides by fourteen so we could bring this down we're going to have 2y is equal to 24 minus 14 which is 10. and now we can divide both sides by 2 and so 10 divided by 2 is 5. so that's we have the ordered pair two comma five d is the correct answer number fourteen the length of a rectangle is 4 more than its width if the area of the rectangle is 60 square feet then what is the perimeter of the rectangle well let's begin by drawing a picture so here we have the length of the rectangle and this is the width now the area of a rectangle is left times width the perimeter of a rectangle is going to be 2l plus 2w it's basically the sum of all four sides our goal is to calculate the perimeter of this rectangle but before we can do that we need to determine the values of l and w so how can we do that let's replace the area with what it's equal to and at 60. so we have 60 is equal to l times w now let's focus on the first sentence of the problem it says the left of a rectangle is 4 more than its width how can we convert that sentence into an equation so the left is 4 more than the width of the rectangle so l is 4 plus w or you can write it as w plus 4. because we have two missing variables we need two equations to solve those two missing variables so what we're going to do is we're going to use substitution these are the two equations that we have we're going to replace l with what it's equal to that is w plus 4. so we're going to have 60 is equal to w plus 4 times w so now we have one equation with one variable now we can solve it but let's distribute w first w times w is a w squared and then w times 4 that's going to be 4w now i'm going to take the 60 and move it to the other side it's positive on the left side but once i move it to the right side it's going to become negative so i have 0 is equal to w squared plus 4w minus 60. now what we can do at this point is we can factor the expression notice that we have a trinomial where the leading coefficient is one so let's find two numbers that multiply to negative 60 but add to four so let's make a list so if we divide negative 60 by 1 we'll get negative 60 but these two numbers don't add up to 4. if we divide negative 60 by 2 it will give us negative 30. if we divide it by 3 we'll get negative 20. if we divide it by 4 that will give us negative 15. if we divided by 5 negative 12 and if we divided by 6 negative 10. now notice that these two they add up to negative four but not positive four so what we could do is switch to sides instead of using positive six and negative ten let's use negative six and positive ten these two they still multiply to negative sixty but now they add up to 4. so we can factor it by writing w minus 6 and w plus 10. so now we can set each factor equal to zero so looking at the first one w minus six is equal to zero if we add six to both sides we can see that w is equal to positive six for the second one if we subtract both sides by ten w is equal to negative ten but when dealing with a real life geometric shape we're not going to have a negative number to represent the length of let's say the the width of the rectangle so this won't apply the width can't be negative 10. that wouldn't make any sense now recall that the length is 4 more than the width of the rectangle so it's going to be 4 plus 6 which means the left of the rectangle is 10. so now let's go back to our picture let's draw the rectangle so it has a length of 10 and a width of 6 so we can see that the area is 10 times 6 which is 60 square feet now this side is 10 and this side is 6. so to calculate the perimeter all we need to do is add up the four sides 10 plus 10 is 20 20 plus 6 is 26 plus another 6 that's 32 so the perimeter of the rectangle is 32 feet and so that's it for this problem number 15 graph the following linear equations so let's start with the first one how can we graph the equation x is equal to two since we only have a variable equal to a number all we need to do is graph a vertical line at x equals two so the graph is just going to look like this because x is always two regardless of whatever y is now let's go ahead and move on to number two let's graph y is equal to three so instead of graphing a vertical line we are going to graph a horizontal line at three so this is the graph y equals three represented in blue now let's move on to the third equation how can we graph the linear equation three x minus four y is equal to twelve notice that it's in standard form ax plus by is equal to c if you want to graph a linear equation in this form the best method that i think you should use is to find the intercepts the x and the y-intercepts so let's begin by calculating the x-intercept to do so set y equal to zero so this is going to be 3x minus 4 times 0 which is 12. so this whole thing becomes 0. so basically you can get rid of it and thus we have 3x is equal to 12 and then we could divide both sides by 3. twelve divided by three is four so this gives us the ordered pair four comma zero so that is the x intercept so let's go ahead and plot it so this is going to be an x-intercept of four now let's determine the y-intercept so instead of replacing y with zero we are going to replace x with zero so three times zero is just zero and so we're gonna have negative four y is equal to twelve next we could divide both sides by negative four so twelve divided by negative four is negative three so we have the point zero negative three so the y-intercept is negative three which is right here and all we need to do at this point is connect these two dots or these two points with a line and so that's how we can graph a linear equation in standard form now let's move on to part four let's graph y equals two x minus three so notice that this equation is in slope intercept form y equals mx plus b so m the slope is 2 and b the y-intercept is negative 3. the first thing i like to do is start with the y intercept which is negative three now all we need in order to graph a linear equation is two points we already have the first one which is the y intercept which is zero negative three we could use the slope to get the next one now the slope is basically the rise over the run the slope is equal to two but you wanna set it up as a fraction two is the same as two over one so two is the rise one is the run so what we're gonna do at this point is we're gonna travel up two units and then one unit to the right and so that's gonna give us the point one negative one now let's repeat the process so let's go up two and then over 1. so the next point is going to be 2 comma 1 and then after that if we do it again it's going to be 3 comma 3 and then connect the dots with a straight line so that's how we can graph a linear equation in slope-intercept form plot the y-intercept and then use the slope to find the next point you