6-1 Solving Systems by Graphing

Quick Review

One way to solve a system of linear equations is by graphing each equation and finding the intersection point of the graph, if one exists.

Example

What is the solution of the system?

y = − 2 x + 2 y = 0.5 x − 3 table with 2 rows and 1 column , row1 column 1 , y equals negative 2 x plus 2 , row2 column 1 , y equals 0.5 x minus 3 , end table

The lines appear to intersect at (2, − 2 ) . negative 2 close .  Check if (2, − 2 ) negative 2 close  makes both equations true.

− 2 = − 2 ( 2 ) + 2 ✓ negative 2 equals negative 2 . open 2 close plus 2 check mark

− 2 = 0 .5 ( 2 ) − 3 ✓ negative 2 , equals 0 , .5 open 2 close minus 3 check mark

So, the solution is (2, − 2 negative 2 ).

Exercises

Solve each system by graphing. Check your answer.

4. y = 3 x + 13 y = x − 3 table with 2 rows and 1 column , row1 column 1 , y equals 3 x plus 13 , row2 column 1 , y equals x minus 3 , end table
5. y = − x + 4 y = 3 x + 12 table with 2 rows and 1 column , row1 column 1 , y equals negative x plus 4 , row2 column 1 , y equals 3 x plus 12 , end table
6. y = 2 x + 3 y = 1 3 x − 2 table with 2 rows and 1 column , row1 column 1 , y equals 2 x plus 3 , row2 column 1 , y equals , 1 third , x minus 2 , end table
7. y = 1.5 x + 2 4.5 x − 3 y = − 9 table with 2 rows and 1 column , row1 column 1 , y equals 1.5 x plus 2 , row2 column 1 , 4.5 x minus 3 y equals negative 9 , end table
8. y = − 2 x − 21 y = x − 7 table with 2 rows and 1 column , row1 column 1 , y equals negative 2 x minus 21 , row2 column 1 , y equals x minus 7 , end table
9. y = x + 1 2 x − 2 y = − 2 table with 2 rows and 1 column , row1 column 1 , y equals x plus 1 , row2 column 1 , 2 x minus 2 y equals negative 2 , end table
10. Songwriting Jay has written 24 songs to date. He writes an average of 6 songs per year. Jenna started writing songs this year and expects to write about 12 songs per year. How many years from now will Jenna have written as many songs as Jay? Write and graph a system of equations to find your answer.
11. Reasoning Describe the graph of a system of equations that has no solution.

6-2 Solving Systems Using Substitution

Quick Review

You can solve a system of equations by solving one equation for one variable and then substituting the expression for that variable into the other equation.

Example

What is the solution of the system?

y = − 1 3 x 3 x + 3 y = − 18 table with 2 rows and 1 column , row1 column 1 , bold italic y equals negative , 1 third , bold italic x , row2 column 1 , 3 bold italic x plus 3 bold italic y equals negative 18 , end table

3 x + 3 y = − 18 Write the second equation. 3 x + 3 ( − 1 3 x ) = − 18 Substitute  − 1 3 x  for  y . 2 x = − 18 Simplify. x = − 9 Solve for  x . y = − 1 3 ( − 9 ) Substitute  − 9  for  x  in the first equation. y = 3 table with 6 rows and 3 columns , row1 column 1 , 3 x plus 3 y , column 2 equals negative 18 , column 3 cap write the second equation. , row2 column 1 , 3 x plus 3 open negative , 1 third , x close , column 2 equals negative 18 , column 3 cap substitute . minus , 1 third , x , for , y . , row3 column 1 , 2 x , column 2 equals negative 18 , column 3 cap simplify. , row4 column 1 , x , column 2 equals negative 9 , column 3 cap solve for . x . , row5 column 1 , y , column 2 equals negative , 1 third , open negative 9 close , column 3 cap substitute . minus 9 , for , x . in the first equation. , row6 column 1 , y , column 2 equals 3 , end table

The solution is ( − 9 , negative 9 comma 3).

Exercises

Solve each system using substitution. Tell whether the system has one solution, infinitely many solutions, or no solution.

12. y = 2 x − 1 2 x + 2 y = 22 table with 2 rows and 1 column , row1 column 1 , y equals 2 x minus 1 , row2 column 1 , 2 x plus 2 y equals 22 , end table
13. − x + y = − 13 3 x − y = 19 table with 2 rows and 1 column , row1 column 1 , negative x plus y equals negative 13 , row2 column 1 , 3 x minus y equals 19 , end table
14. 2 x + y = − 12 − 4 x − 2 y = 30 table with 2 rows and 1 column , row1 column 1 , 2 x plus y equals negative 12 , row2 column 1 , negative 4 x minus 2 y equals 30 , end table
15. 1 3 y = 7 3 x + 5 3 x − 3 y = 5 table with 2 rows and 1 column , row1 column 1 , 1 third , y equals , 7 thirds , x plus , 5 thirds , row2 column 1 , x minus 3 y equals 5 , end table
16. y = x − 7 3 x − 3 y = 21 table with 2 rows and 1 column , row1 column 1 , y equals x minus 7 , row2 column 1 , 3 x minus 3 y equals 21 , end table
17. 3 x + y = − 13 − 2 x + 5 y = − 54 table with 2 rows and 1 column , row1 column 1 , 3 x plus y equals negative 13 , row2 column 1 , negative 2 x plus 5 y equals negative 54 , end table
18. Business The owner of a hair salon charges $20 more per haircut than the assistant. Yesterday the assistant gave 12 haircuts. The owner gave 6 haircuts. The total earnings from haircuts were $750. How much does the owner charge for a haircut? Solve by writing and solving a system of equations.

Table of Contents