SOLVING SYSTEMS OF LINEAR EQUATIONS BY SUBSTITUTION WORKSHEET

Problems1-5 : Solve the given system of linear equations by substitution.

Problem 1 :

y = 2x

x + y = 9

Problem 2 :

x = 4y

2x - 7y - 5 = 0

Problem 3 :

y = 6x - 11

-2x - 3y + 7 = 0

Problem 4 :

x = 1 - 3y

x + y = 5

Problem 5 :

-3x + 2y = 4

-x + y = 3

Problem 6 :

x + 3y = -5

2x - 5y = 12

Problem 7 :

-3x + 3y = 4

x - y = -3

Problem 8 :

6x - 2y = -5

y = 3x + 2.5

Problem 9 :

There are two numbers such that two times the first numbers is equal to 3 less than three times the second number. If the difference between the first number and two times the second number is equal to -9, find the numbers.

Problem 10 :

In a magic show, the cost of an adult ticket is \$12 and that of a child ticket is \$8. Find the number adults tickets and kids tickets sold, if a total of 500 tickets were sold for a total of \$4800.

y = 2x ----(1)

x + y = 9 ----(2)

Substitute y = 2x into (2).

x + 2x = 9

3x = 9

Divide both sides by 3.

x = 3

Substitute x = 3 into (1).

y = 2(3)

y = 6

The solution is (3, 6).

x = 4y ----(1)

2x - 7y - 5 = 0 ----(2)

Substitute x = 4y into (2).

2(4y) - 7y - 5 = 0

8y - 7y - 5 = 0

y - 5 = 0

y = 5

Substitute y = 5 into (1).

x = 4(5)

x = 20

The solution is (20, 5).

y = 6x - 11 ----(1)

-2x - 3y + 7 = 0 ----(2)

Substitute y = 6x - 11 into (2).

-2x - 3(6x - 11) + 7 = 0

-2x - 18x + 33 + 7 = 0

-20x + 40 = 0

Subtract 40 from both sides.

-20x = -40

Divide both sides by -20.

x = 2

Substitute x = 2 into (1).

y = 6(2) - 11

y = 12 - 11

y = 1

The solution is (2, 1).

x = 1 - 3y ----(1)

x + y = 5 ----(2)

Substitute x = 1 - 3y into (2).

(1 - 3y) + y = 5

1 - 3y + y = 5

1 - 2y = 5

Subtract 1 from both sides.

-2y = 4

Divide both sides by -2.

y = -2

Substitute y = -2 into (1).

x = 1 - 3(-2)

x = 1 + 6

x = 7

The solution is (7, -2).

-3x + 2y = 4 ----(1)

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

Solve (2) for y.

-x + y = 3

y = x + 3 ----(3)

Substitute y = x + 3 into (1).

-3x + 2(x + 3) = 4

-3x + 2x + 6 = 4

-x + 6 = 4

Subtract 6 from both sides.

-x = -2

Multiply both sides by -1.

x = 2

Substitute x = 2 into (3).

y = 2 + 3

y = 5

The solution is (2, 5).

x + 3y = -5 ----(1)

2x - 5y = 12 ----(2)

Solve (1) for x.

x + 3y = -5

Subtract 3y from both sides.

x = -5 - 3y ----(3)

Substitute x = -5 - 3y into (2).

2(-5 - 3y) - 5y = 12

-10 - 6y - 5y = 12

-10 - 11y = 12

-11y = 22

Divide both sides by -11.

y = -2

Substitute y = -2 into (3).

x = -5 - 3(-2)

x = -5 + 6

x = 1

The solution is (1, -2).

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

x - y = -3 ----(2)

Solve (2) for x.

x - y = -3

x = y + 3

Substitute x = y + 3 into (1).

-3(y + 3) + 3y = 4

-3y - 9 + 3y = 4

-9 = 4 (false)

In the above step of solving the given system, the variable y is no more and '-9 = 4' is false.

So, the given system of linear equations has NO solution.

6x - 2y = -5 ----(1)

y = 3x + 2.5 ----(2)

Substitute y = 3x + 2.5 into (1).

6x - 2(3x + 2.5) = -5

6x - 6x - 5 = -5

-5 = -5 (true)

In the above step of solving the given system, the variable x is no more and '-5 = -5' is true.

So, the given system of linear equations has infinitely many solutions.

Let x be the first number and y be the second number.

From the given information,

2x = 3y - 10 ----(1)

x - 2y = -9 ----(2)

Multiply both sides of (2) by 2.

2(x - 2y) = 2(-9)

2x - 4y = -18 ----(3)

Substitute 2x = 3y - 10 into (3).

(3y - 10) - 4y = -18

3y - 10 - 4y = -18

-y - 10 = -18

-y = -8

Multiply both sides by -1.

y = 8

Substitute y = 8 into (2).

x - 2(8) = -9

x - 16 = -9

x = 7

The numbers are 7 and 8.

Let x be the number of adult tickets and y be the number of child tickets.

Given : Total number of tickets sold is 500.

x + y = 500 ----(1)

Given : Total amount collected \$4800.

12x + 8y = 4800

Divide both sides by 4.

3x + 2y = 1200 ----(2)

Solve (1) for y.

x + y = 500

Subtract x from both sides.

y = 500 - x

Substitute y = 500 - x into (2).

3x + 2(500 - x) = 1200

3x + 1000 - 2x = 1200

x + 1000 = 1200

Subtract 1000 from both sides.

x = 200

Substitute x = 200 into (1).

200 + y = 500

Subtract 200 from both sides.

y = 300

Number of adult tickets sold is 200 and that of kid tickets is 300.

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