consider the following two equations two x plus three y is equal to eight and also five x minus three y is equal to negative one how can we solve these two equations using elimination which is also known as the addition method using elimination what you want to do is add the two equations 3y plus negative 3y these two cancel 2x plus 5x is 7x and 8 plus negative 1 is 7. so now if we divide both sides by seven we can get the value of x seven divided by seven is one now once you have the value of x you can plug it into the first or even the second equation let's use the first one let's replace x with one so we're gonna have two times one plus three y is equal to eight two times one is two next let's subtract both sides by two so that cancels so we're going to have now is 3y is equal to 8 minus 2 which is 6. and now let's divide both sides by 3. 6 divided by three is two so y equals two so to write your answer as an ordered pair it's gonna be one comma two now let's try another example let's say that 2x plus 5y is equal to 19 and also x minus 2y is equal to negative 4. now this example is a little different if we add the two equations x or y will not cancel completely so we need to modify it so we need to choose if we want to cancel x or y since x has smaller numbers let's try to cancel x all we need to do is multiply the second equation by negative two since we have a positive two here we want this to be a negative two so that it's going to cancel now i'm going to rewrite the first equation now the second equation multiply everything times 2 and 1x times negative 2 is negative 2x negative 2y times negative 2 is positive 4y and negative 4 times negative 2 is positive 8. now let's add the two equations so those two will cancel 5y plus 4y is 9y 19 plus 8 is 27. now let's divide both sides by 9. 27 divided by 9 is 3. now let's find the value of x so let's use the first equation let's replace y with three five times three is fifteen next let's subtract both sides by 15. so those will cancel 19 minus 15 that's equal to 4 and if we divide both sides by 2 we can see that 4 divided by 2 is 2 so x is equal to 2. so therefore the solution to this problem is 2 comma 3. that's the answer so that's how you can solve a system of two equations using the elimination method in this lesson we're going to talk about how to solve a system of two equations using substitution so the first equation that we have is y is equal to 5 minus 2x and the second one is 4x plus 3y is equal to 13. so what should we do here if we want to solve by substitution well what we want to do is we want to replace or substitute y in the second equation with five minus two x because y is equal to five minus two x so we're gonna have is four x plus three times five minus two x and that's equal to thirteen now let's go ahead and distribute three so three times five that's fifteen and three times negative 2x is negative 6x in our next step we need to combine like terms 4x plus negative 6x that's negative 2x so negative 2x plus 15 is equal to 13. next let's subtract both sides by negative 13 or by 13 rather so actually i want to subtract it by 15 not by 13. that's what i meant to say so this will cancel negative 2x is equal to 13 minus 15 which is negative 2. now let's divide both sides by negative two negative two divided by itself is equal to one so x is one now that we have the value of x let's plug into the first equation to get the value of y so five minus two times one two times one is two and five minus two is three so y is equal to three therefore the solution to this system of equations is one comma three now what about this example let's say y is equal to three x plus two and also y is equal to seven x minus six how can we use substitution to solve for x and y well we can replace this y with three x plus two because y equals three x plus two so three x plus two is therefore equal to seven x minus six and now let's find a value of x so let's begin by adding six to both sides and simultaneously let's add or let's subtract both sides by three x negative six plus six is zero three x minus three x is zero on the left two plus six is eight and on the right seven x minus three x is four x 4x next let's divide both sides by 4. 8 divided by 4 is 2 so x is equal to 2. now let's plug it into the first equation you can also plug it into the second equation three times two plus two three times two is six six plus two is eight if you plug it into the second equation it's going to be seven times two minus six seven times two is fourteen fourteen minus six is eight so you get the same answer so y equals eight so the answer is two comma eight as an ordered pair that's the solution to the equation now here's another example 4x plus 2y is equal to 14 and also uh 3x minus 5y let's say that's equal to negative 22. what can we do to solve for x and y how can we use substitution to get the answers now typically when x and y are lying like the way you see it i prefer to use the elimination method but sometimes you may be given a problem like this and you need to use the substitution method what you want to do is solve for x or y i think it's easier to solve for y in the first equation so let's do that so i'm going to rewrite the equation here let's begin by subtracting both sides by 4x which is equivalent to taking the 4x moving it from the left side to the right side on the left side 4x is positive on the right side it's going to be negative so therefore 2y is equal to negative 4x plus 14. now i'm going to divide both sides by 2 or basically everything by 2. so y is equal to negative 2x plus 7. so now that i have y by itself in the second equation i'm going to replace y with negative two x plus seven so let's substitute y with negative two x plus seven so three x minus five times negative two x plus 7 that's going to equal negative 22. now let's distribute negative 5. negative 5 times negative 2x that's positive 10x negative 5 times positive 7 that's equal to negative 35. so now let's combine like terms 3x and 10x that's 13x now in the next step let's add 35 to both sides negative 22 plus 35 is 13. so 13x equals 13. now let's divide both sides by 13. we can see that x is equal to one now the next thing we need to do is take that value plug it into that equation to get y so negative two times one plus seven negative two times one is negative two negative two plus seven is five so y is equal to five therefore the solution as an ordered pair is one comma five that's the answer you