in this video i'm going to go over some basic questions relating to linear functions so let's start with this problem what is the slope of the line passing through the points 2 comma negative 3 and four comma five so to calculate the slope between two points you need to use this formula it's y two minus y one divided by x two minus x one the first number is x the second one is y so let's call two negative three x one y one and four comma five x two y two so y two in this example is five y one is negative three x two is four x 1 is 2. now 5 minus negative 3 is the same as 5 plus 3 and 5 plus 3 is 8 4 minus 2 is 2 and 8 divided by 2 is 4. so that's the slope that passes through these two points number two what is the slope and y-intercept of the line y equals two x minus three now you need to know that this equation is in slope intercept form which is y equals mx plus b so m represents the slope and b is the y-intercept so we could see that the slope is equal to 2 and the y-intercept is negative 3. and so that's it for this question now the y-intercept you can write it as an ordered pair if you want you could say it's zero negative three or simply the y-intercept is equal to negative three you can write it both ways but that's it for this problem number three graph the equations x equals two and y equals three so let's start with x equals two how can we graph this particular equation now whenever x is equal to a number the type of graph you're going to get it's going to be a line but specifically a vertical line at x equals 2. now for the other one y equals 3 the graph is simply a horizontal line where y is string and that's it that's all you need to do in order to graph these two equations number four graph the equation using the slope-intercept method so we have the equation y is equal to three x minus two so we could see that the slope is equal to three and a y-intercept is negative two i'm just going to write the ordered pair so we have the point zero negative two so here's the first point at zero negative two now the slope is 3 which represents the rise over the run so we need to rise three units and go over one unit to the right to get the next point so the next point is going to be at one comma one and then we could do it again rise three over one so the next point is going to be two comma four and then we can just connect these points with a straight line and so that's a rough estimate of the graph y equals three x minus two number five graph the equation two x minus three y is equal to six using the x and y-intercepts now to find the x-intercept replace y with zero negative three times zero is simply zero and so zero is nothing so this just disappears so 2x is equal to 6 and if we divide both sides by 2 we can see that x is equal to 3. so that's the x-intercept which means we have the point 3 comma 0. now to calculate the y-intercept we need to replace x with zero and solve for y so two times zero is nothing and so we have negative three y is equal to six and now let's divide both sides by negative three so then positive six excuse me divided by negative three that's negative two and so that's the y intercept so we have the point zero negative two at this point we can make the graph so the first point is three zero that's the x-intercept the next one is zero negative two that's the y-intercept and then just connect these two points with a straight line and it appears i missed it so there it is so that's how you can graph a linear equation in standard form there's three forms you need to be familiar with this is known as slope intercept form this is called standard form and then this equation is the point-slope form number six write the equation of the line passing through the point two comma five with a slope of three so whenever you're given the point and a slope it's best to use the point-slope formula to write the linear equation so x one is two y one is five and m is three so it's going to be y minus five is equal to three times x minus two so that's the linear equation in point slope form now let's convert it to slope intercept form so i'm going to distribute the three three times x that's three x and then it's three times negative two which is negative six so now let's add five to both sides negative six plus five is negative one so this is the linear equation in slope intercept form and that's the answer number seven write the equation of the line passing through the points negative three comma one and two comma negative four now we can't use the point slope formula yet because i don't have the slope so we need to calculate the slope first so it's y2 minus y1 divided by x2 minus x1 so this is going to be x1 and that's going to be y1 and this is x2 and y2 so y2 is negative 4 y1 is positive 1. x2 is 2 x one is negative three so negative four minus one is negative five two minus negative three is the same as two plus three that's five negative five divided by five is negative one so that's the slope it's equal to negative one now we can use the point slope formula so i'm going to use the first point negative three comma one so y minus y one is equal to m times x minus x one so y one is one m is negative one and x one is negative three so this is y minus one equals to negative x negative one times x plus three these two negative signs will become positive now let's distribute the negative one so it's going to be negative x minus three and then let's add one to both sides so y is equal to negative x and negative three plus one is negative two so this is the final answer it's negative x minus two number eight write the equation of the line passing through the point three negative two and parallel to the line two x plus five y minus three so what you need to know is that parallel lines have the same slope so if the slope here is two then the slope of the other line will also be 2. so we already have the point passing through or that's part of that line we need to find the slope of this line which will be the slope of the line of the equation that we're looking to find so let's turn this equation and change it into its slope intercept form so first i'm going to move the 3 to this side so it's going to be positive 3 on the right side and then i'm going to take the 2x move it to that side where it's going to be negative 2x so i have 5y is equal to negative 2x plus 3 and then divide every term by 5. so y is equal to negative two over five times x plus three over five so the slope of this line is the number in front of x when it's in slope intercept form so the slope is negative two over five so now i can use the point slope formula to write the equation of the line so this is going to be x1 and y1 so y one that's negative two m is negative two over five x one is three so this is gonna be y plus 2 and then let's distribute the negative 2 over 5 to x minus 3. now negative 2 over 5 times negative 3 negative 2 times negative 3 is 6. so this is going to be 6 over 5. and now let's subtract both sides by two now two over one i'm going to multiply this by five over five to get common denominators so negative 2 is the same as negative 10 over 5. now 6 minus 10 is 4. so this is going to be negative 4 over 5. and this is the answer so that's the equation of the line in slope-intercept form number nine write the equation of the line passing through the point negative four negative three and perpendicular to the line three x minus four y plus five so first we need to find the slope of this equation so let's get y by itself i'm going to move the 3x and the 5 to the other side so it's going to be negative 3x and negative 5 on the right side now i need to divide each term by negative four so y is equal to three over four times x plus five over four whenever you divide two negative numbers you're going to get a positive result so the slope of this line is 3 over 4. now let's say if we have two lines line k and line out and let's say that these two lines are perpendicular which means that they meet at right angles or at a 90 degree angle now let's say that the slope of line k is 2 over 5. what do you think is the slope of line l perpendicular lines have a slope that is the negative reciprocal of each other so this slope is positive the other one will be negative and then you need to flip the fraction so the reciprocal of two over five is five over two and so that tells us that these two lines are perpendicular so the slope for this equation is three over four so therefore the slope of the perpendicular line that we want to find is going to be the negative reciprocal of that fraction which is negative four divided by three now let's finish this problem so let's use the point slope formula so x1 is going to be negative 4 y1 is negative 3. so it's going to be y minus negative 3 and the slope is negative 4 over 3 and then x1 is negative 4. so this becomes y plus 3 and that's equal to negative 4 divided by 3 times x plus 4. so that's the answer in point slope form now let's distribute the negative 4 over 3. let's convert it to slope intercept form so this is going to be negative 4 over 3 times x and then negative 4 times 4 is negative 16. now let's subtract both sides by three now we need to get common denominators so i'm going to multiply this 3 by 3 over 3 so it becomes negative 9 over 3. negative 16 minus 9 is negative twenty five so this is the final answer it's negative four over three x minus twenty five over three you