5.2 Solving Systems of Linear Equations by Substitution

Solving Linear Systems by Substitution

Example 1:Solving a System of Linear Equations by Substitution

Solve the system of linear equations by substitution.y = -2x – 9Equation 16x – 5y = -19Equation 2Step 1: Equation 1 is already solved for y.

Step 2: Substitute -2x – 9 in for y in Equation 2 and solve for x.

6x – 5(-2x – 9) = -19 6x + 10x + 45 = -19 16x + 45 = -19 - 45 - 45 16x = -64 x = -4

Step 3: Substitute -4 in for x in Equation 1 and solve or y.

y = -2(-4) – 9y = 8 – 9y = -1

The solution is (-4,-1).

Remember to check your work by substituting the values back in the original equations!

You try!

Solve the system of linear equations by substitution. Check your solution!

1) y = 3x + 14 2) x = 6y – 7

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

(-2,8) (-1,1)

Example 2:Solving a System of Linear Equations by SubstitutionSolve the system of linear equations by substitution.-x + y = 3Equation 13x + y = -1Equation 2Step 1: Solve Equation 1 for y.

y = x +3

Step 2: Substitute x + 3 in for y in Equation 2 and solve for x.

3x + (x + 3) = -1 4x + 3 = -1 -3 -3

4x = -4 x = -1

Step 3: Substitute -1 in for x in Equation 1 and solve or y.

-(-1) + y = 3 1 + y = 3 y = 2

The solution is (-1,2).

Remember to check your work by substituting the values back in the original equations!

You try!

3) –x + y = -4

4x – y = 10

(2,-2)

Example 3:Solving Real-Life ProblemsA drama club earns $1040 from a production. A total of 64 adult tickets and 132 student tickets are sold. An adult ticket cost twice as much as a student ticket. Write a system of linear equations to represent the situation. What is the cost of each ticket type?

Let x be the price (in dollars) of an adult ticket.Let y be the price (in dollars) of a student ticket.

x = 2y

Equation 1 x = 2y Equation 2

Step 1: Equation 2 is already solved for x.

Step 2: Substitute 2y in for x in Equation 1

and solve for y.

64(2y) + 132y = 1040

128y + 132y = 1040

260y = 1040

y = 4

Step 3: Substitute 4 in for y in Equation 2 and solve or x.

x = 2(4) x = 8

The solution is (8,4). This means that an adult ticket cost $8 and a student ticket cost $4.

Remember to check your work by substituting the values back in the original equations!

You try!

4) There are a total of 64 students in a drama club and a yearbook club. The drama club has 10 more students than the yearbook club. Write a system of linear equations that represents this situation. How many students are in each club?

Let x be the number of students in the drama club.Let y be the number of students in the yearbook club.

x -10 = yThe solution is (37,27). This means that there are 37 students in the drama club and 27 students in the yearbook club.