Name: ____________________________________
Date: __________________
Solving Linear Systems by Substitution
In order to solve any system, we must find all coordinate pairs  x, y  that satisfy both equations, or as
we have learned before, make both equations true. You have already learned how to solve a system of
equations graphically. We now explore the first way to do this algebraically – by substitution. This
method relies on a very important principal:
SUBSTITUTION PROPERTY OF EQUALITY
EQUALS MAY BE SUBSTITUTED FOR EQUALS – one equal quantity may replace another in an equation.
Exercise #1: Consider the system of equations given to the right:
y  2x 1
y  x  5
(a) Solve this system graphically using the grid provided.
y
(b) Solve this system by substituting the value of y from the first
equation into the second.
Exercise #2: Find the intersection point of the two lines whose equations are given below. Only an
algebraic solution is acceptable.
y  8 x  12
y  3x  23
x
The method of substitution is particularly useful in solving systems when one of the original equations
is solved for x or y or if one of the equations can easily be solved for either x or y.
Exercise #3: Solve the following system by substitution.
2 x  3 y  16
x  2 y  6
Exercise #4: Solve the following system by substitution.
4 x  3 y  29
y  3x  18
Exercise #5: Consider the system of equations given at the right:
(a) Do each of these lines pass through the two points (4, -1) and (0, 1)?
2x  4 y  4
2  2y  x
(b) What does your answer in part (a) tell you about the graphs of the two given lines?
(c) Solve the system algebraically and explain your answer.
Name: ____________________________________
Date: __________________
Solving Linear Systems by Substitution
Homework
Skills
1. Which of the following is the solution to the system shown to the right?
(1)  2,11
(3)  4, 3
(2) 1,  3
(4)  0, 5
4 x  y  19
y  2x  5
2. Which of the following is the point of intersection of the lines whose equations are
1
y   x  3 and y  2 x  8 ?
2
(1)  2, 4
(3)  0, 8
(2)  4,1
(4)  7,  2 
3. Solve each of the following systems algebraically by using substitution.
(a) x  3 y  3
y  3x  7
(c)
x y 3
3x  3 y  5
(b) y  5 x  12
y  2 x  16
(d) 3x  2 y  0
3x  y  18
4. Algebraically, find the intersection points of the two lines given in each part of this problem.
(a) y  3x  10
y  6 x  25
(b) y  0.12 x  18
y  0.08 x  25
Reasoning
5. Consider the system of equations shown at the right:
y  2x 1
y  2x  4
(a) Solve the system of equations by graphing both on the axes to
the right.
y
(b) Solve the system by using substitution.
(c) How do your answers from part (a) and (b) support one another? Explain.
6. Luke believes that the point  3,  2  is a solution to the system of equations below. Is Luke
correct? Justify your answer.
3x  2 y  5
2x  y  8
x