1. Solving systems of equations
Substitution method

2. Solve the following system using substitution
−7𝑥−3𝑦=21
−8𝑥+6𝑦=24

3. Step 1: Solve one of the equations for one of the variables
It doesn’t matter which equation or variable you choose, although sometimes one may be easier to solve for than the other.
Ex.1) −7𝑥−3𝑦=21
−8𝑥+6𝑦=24
Let’s Solve this equation for y

4. −8𝑥+6𝑦=24
+8𝑥 𝑥
6𝑦 = 8𝑥
+24
𝑦= 4 3 𝑥+4
We now have an expression in terms of x that is equal to y.

5. −7𝑥−3𝑦=21 𝑦= 4 3 𝑥+4 ( 4 3 𝑥+4) −8𝑥+6𝑦=24 =21 −7𝑥−3 𝑦
We can replace the second equation in our system with the new one where we have solved for y.
We can then substitute for y in the first equation. REPLACING y with the expression it is equal to.
=21
−7𝑥−3 𝑦

6. Now that we have substituted in for y, we have a new equation that only has the variable x in it. We can now solve this for x.
−7𝑥− 𝑥+4 =21
Use the distributive property to take care of the parenthesis
−7𝑥− 𝑥− 3 4 =21
−7𝑥−4𝑥−12=21

7. −7𝑥−4𝑥−12=21 −11𝑥−12=21 −11𝑥−12=21 +12 +12 −11𝑥 =33 −11 −11 𝑥=−3
Combine any like terms
−11𝑥−12=21
Finish solving the resulting two step equation
−11𝑥−12=21
−11𝑥 =33
−11
−11
𝑥=−3

8. Once we have solved for one variable, in this case x, we can substitute the value of x into one of the original equations to find y. We can also use any equation we derived from the original equations.
𝑦= 4 3 𝑥 +4 −3
Simplify the resulting equation
𝑦=−4+4
𝑦=0

9. −3,0 Don’t forget to write your solution as an ordered pair.
And also remember you can check your solution by substituting both x and y into the two original equations and verifying that the resulting equations are true.

10. Summary Solve one of the equations for one of the variables (y)
Substitute the value of the variable into the other equation
Solve the resulting equation for the second variable (x)
Substitute the numerical value of the second variable (x) into either equation to find the numerical value of the first variable (y)
Write your answer as an ordered pair (x,y)
Check your answer by substituting for both variables and making sure the resulting equations are true.