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!
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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.