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.
1. Answer :
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).
2. Answer :
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
Add 3 to both sides.
y = 5
Substitute y = 5 into (1).
x = 4(5)
x = 20
The solution is (20, 5).
3. Answer :
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).
4. Answer :
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).
5. Answer :
-3x + 2y = 4 ----(1)
-x + y = 3 ----(2)
Solve (2) for y.
-x + y = 3
Add x to both sides.
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).
6. Answer :
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
Add 10 to both sides.
-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).
7. Answer :
-3x + 3y = 4 ----(1)
x - y = -3 ----(2)
Solve (2) for x.
x - y = -3
Add y to both sides.
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.
8. Answer :
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.
9. Answer :
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
Add 10 to both sides.
-y = -8
Multiply both sides by -1.
y = 8
Substitute y = 8 into (2).
x - 2(8) = -9
x - 16 = -9
Add 16 to both sides.
x = 7
The numbers are 7 and 8.
10. Answer :
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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