1. SOLVING LINEAR SYSTEMS WITH SUBSTITUTION by Sam Callahan

2. By now you’ve learned to solve systems of equations using graphing and finding where the lines intersect:

3. The problem with solving by graphing though, is evident when you look at graphs like the one below. This solution (the blue point where the lines intersect) isn’t on a gridline and very hard to accurately identify

4. Although graphing is simple and visual, it is really only accurate enough to use with systems that have integer answers. Substitution is a way to solve systems of equations analytically, or without graphing.

5. If we take the same system of equations: We can make solving these equations possible by working with one variable at a time. To do this, we substitute one side of the equation in for the other variable.

6. If we know that y = x+3, we can plug in (x+3) in for y in the second equation to find out what x is. 3x – y = 7substitute (x+3) for y 3x – (x+3) = 7distribute -1 through the parentheses 3x – x – 3 = 7simplify 2x – 3 = 7simplify 2x = 10simplify x=5

7. Now that we know what x is (5), we can plug 5 in for x in either equation to find out what y is. I like to use whichever equation has simpler numbers to work with. In this case, that equation is:

8. If x = 5 y = x + 3 y = 5 + 3 y = 8

9. x = 5 and y = 8, so our solution to the system In (x, y) form is (5, 8) Now let’s check our answer.

10. Checking your solution You should check your answer using the equation that you didn’t just solve. For example, my last step was plugging in 5 for x into y = x + 3 I should check with the other equation, 3x – y = 7

11. Checking your solution x = 5y = 8 3x – y = 7 plug in your values for x and y 3(5) – (8) = 7 simplify 15 – 8 = 7 simplify 7 = 7 make sure your statement is true We ended up with a true statement, so our solution works!

12. Try this one… 4x – 12y = 20 3x + 9y = 45

13. Try this one… 4x – 12y = 20 3x + 9y = 45 Unlike the previous example, we aren’t given an equation right away that says what x or y is equal to, so we have to simplify one of these equations so that it reads y=_____ or x=______ Choose one of the equations to simplify. I’ll use 3x + 9y = 45

14. 3x + 9y = 45 You can start with either variable, but I want to solve for x first because I don’t want a fraction that would result if I divided everything by 9. 3x + 9y = 45 subtract 9y to put it on the right side of the equation 3x = 45 – 9y divide everything by 3 so that x will be on its own /3 /3 /3 x = 15 – 3y

15. Now we have what x is equal to, so we can plug in “15 – 3y” for x in the other equation. 4x – 12y = 20 plug in the expression for x 4(15 – 3y) – 12y = 20 distribute 60 – 12y – 12y = 20 simplify 60 – 24y = 20 simplify 40 = 24y simplify y = 5/3 I used a calculator for this step, but you could also simplify this fraction (40/24) by hand

16. y = 5/3 Plug this y-value into the other equation we found (x = 15 – 3y) to find x. x = 15 – 3yplug in (5/3) for y x = 15 – 3(5/3)simplify x = 15 – 53 times (5/3) is 5 x = 10

17. x = 10y = 5/3 So our solution is (10, 5/3) Always check your solution!

18. To review, 4x – 12y = 20 3x + 9y = 45  We simplified one of the equations 3x + 9y = 45  x = 15 – 3y  Plugged this “15 – 3y” in for x in the other equation 4x – 12y = 20 4(15 – 3y) – 12y = 20  Solved for y y = 5/8  Plugged in this y-value into the other equation to find x x = 15 – 3(5/3) x = 10