Solving Systems of

Equations by Substitution

byTammy Wallace

Varina High School

Solving by Systems by Substitution

Another method for solving system of equations is the substitution method.

This is done by replacing one variable (y) with an equivalent expression that contains the other variable (x).

From there, a one-variable equation is created, that will be used to find the solution set.

Find the solution set for Procedures

Solve one equation for one variable.  NOTE: Either equation can be chosen

Which equation did you choose?

______________________ 

       

 

While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.  What did that variable equal?  ______________________ 

  

 

Because this equation is already solved for y, it is easier to use this one.

7x + 2y = 37  7x + 2(3x - 1) = 37 7x + 6x – 2 = 37 13x – 2 = 37 + 2 + 2 13x = 39 13 13 x = 3

x=3

Find the solution set for Procedures

Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal?  ______________________

Remember x = 3y = 3x – 1 y = 3(3) – 1 y = 9 – 1 y = 8y=8

Find the solution set for Procedures 

a) What is/are the solutions to the system?

b) If graphed, what type of lines would this system form and how can you determine this WITHOUT graphing the system?  

c) Solve both equations for y and graphing with your calculator. Is the solution set correct?

 

If x = 3 and y = 8, the solution set is (3, 8)

The graph would form intersecting lines because there is ONE SOLUTION to the system.

YES!

Find the solution set for Procedures

Solve one equation for one variable.  NOTE: Either equation can be chosen

Which equation did you choose?

______________________ 

  

     

 

While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.  What did that variable equal?  ______________________ 

  

 

-x – 2y = 8 -x – 2(-4x + 3) = 8 -x + 8x – 6 = 8 7x – 6 = 8 + 6 +6 7x = 14 7 7 x = 2

x=2

4x + y = 3 -4x -4x y = -4x + 3

Procedures

Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal?  ______________________y=−5

Find the solution set for

Remember x = 2-x – 2y = 8 -2 – 2y = 8 +2 +2 -2y = 10 -2 -2 y = -5

Procedures 

a) What is/are the solutions to the system?

b) If graphed, what type of lines would this system form and how can you determine this WITHOUT graphing the system?  

c) Solve both equations for y and graphing with your calculator. Is the solution set correct?

 

If x = 2 and y = -5, the solution set is (2, -5_)

The graph would form intersecting lines because there is ONE SOLUTION to the system.

YES!

Find the solution set for

Find the solution set for Procedures

Solve one equation for one variable.  NOTE: Either equation can be chosen

Which equation did you choose?

______________________ 

  

     

 

While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.  What did that variable equal?  ______________________ 

  

 

4x + y = 3 4(-2y – 8) + y = 3 -8y – 32 + y = 3 -7y – 32 = 3 +32 +32 -7y = 35 -7 -7 y = -5

𝑦=−5

-x – 2y = 8 + 2y = +2y -x = 2y + 8 x = -2y - 8

-1( )

Procedures

Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal?  ______________________𝑥=2

Find the solution set for

Remember y = -54x + y = 3 4x + (-5) = 3 4x – 5 = 3 + 5 +5 4x = 8 4 4 x = 2

Procedures 

a) What is/are the solutions to the system?

b) What do you notice about the solution to this system and solution to problem # 2? What can you conclude about the process used their final answers?

 

If x = 2 and y = -5, the solution set is (2, -5_)

The solutions are the same. It doesn’t matter which equation you solve for first and what variable you solve for first. As long as the procedure is done correctly, the solution will be the same.

Find the solution set for

Find the solution set for Procedures

Solve one equation for one variable.  NOTE: Either equation can be chosen

Which equation did you choose?

______________________ 

  

     

 

While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.  

  

 

What happened when solving the equation for y?

What can we conclude about the solution?

What type of lines will the graph be?

x + y = 7 3 – y + y = 7 3 = 7

The variable cancelled leaving 3 = 7.

Because this equation is already solved for y, it is easier to use this one.

Because 3 can NEVER equal 7, there are NO SOLUTIONS.

The lines are parallel.

Find the solution set for Procedures

5) Solve the system of equation

   

 

While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.   

  

What happened when solving the equation for y?

What can we conclude about the solution?

What type of lines will the graph be?

2x + y = 4-2x -2xy = -2x + 44x + 2y = 8 4x + 2(-2x + 4) = 8 4x – 4x + 8 = 8 8 = 8 Both sides of the equation are

equal

There are infinite many solutions.

The lines are coinciding.