Problems 1-7 :Solve each of the following systems of linear equations by elimination.
Problem 1 :
x + y = 2
x - y = 0
Problem 2 :
-3x + 2y = -2
3x - y = -8
Problem 3 :
3x + 5y = 5
x + 5y = -5
Problem 4 :
-2x + y = 14
4x - 3y = -32
Problem 5 :
5x - 6y = 1
7x + 8y = 1
Problem 6 :
2x - 3y = 3
-2x + 3y = 1
Problem 7 :
x - 2y = -5
-7x + 14y = 35
Problem 8 :
Find two numbers such that the sum of them is 18 and the difference between them is 2.
1. Answer :
x + y = 2 ----A
x - y = 0 ----B
Since we find y in A and -y in B, by adding A and B, y can be eliminated.
A + B :
(x + y) + (x - y) = 2 + 0
x + y + x - y = 2
2x = 2
Divide both sides by 2.
x = 1
Substitute x = 1 into A.
1 + y = 2
Subtract 1 from both sides.
y = 1
The solution is (1, 1).
2. Answer :
-3x + 4y = 6 ----A
3x - y = 3 ----B
Since we find -3x in A and 3x in B, by adding (1) and (2), x can be eliminated.
A + B :
(-3x + 4y) + (3x - y) = 6 + 3
-3x + 4y + 3x - y = 9
3y = 9
Divide botyh sides by 3.
y = 3
Substitute y = 3 into B.
3x - 3 = 3
Add 3 to both sides.
3x = 6
Divide both sides by 3.
x = 2
The solution is (2, 3).
3. Answer :
3x + 5y = 5 ----A
x + 5y = -5 ----B
Multiply both sides of A by -1.
-1(3x + 5y) = -1(5)
-3x - 5y = -5 ----C
Add B and C to eliminate y.
B + C :
(x + 5y) + (-3x - 5y) = (-5) + (-5)
x + 5y - 3x - 5y = -5 - 5
-2x = -10
Divide both sides by -2.
x = 5
Substitute x = 5 into (B).
5 + 5y = -5
Subtract 5 from both sides.
5y = -10
Divide both sides by 5.
y = -2
The solution is (5, -2).
4. Answer :
-2x + y = 14 ----A
4x - 3y = -32 ----B
Multiply A by 2.
2(-2x + y) = 2(14)
-4x + 2y = 28 ----C
Add B and C to eliminate x.
B + C :
(4x - 3y) + (-4x + 2y) = -32 + 28
4x - 3y - 4x + 2y = -4
-y = -4
Multiply both sides by -1.
y = 4.
Substitute y = 4 into A.
-2x + 4 = 14
Subtract 4 from both sides.
-2x = 10
Divide both sides by -2.
x = -5
The solution is (-5, 4).
5. Answer :
5x - 6y = -11 ----A
7x + 8y = 1 ----B
In the above system of linear equations, the number in front of either the variable x or y is not same. We have -6 infront of y in A and 8 in B.
Least common multiple of (6, 8) = 24.
Multiply A by 4 to get -24 infront of y and multiply B by 3 to get 24 infornt of y.
Multiply A by 4.
4(5x - 6y) = 4(-11)
20x - 24y = -44 ----C
Multiply B by 3.
3(7x + 8y) = 3(1)
21x + 24y = 3 ----D
Add C and D to eliminate.
C + D :
(20x - 24y) + (21x + 24y) = -44 + 3
20x - 24y + 21x + 24y = -41
41x = -41
Divide both sides by 41.
x = -1
Substitute x = -1 into B.
7(-1) + 8y = 1
-7 + 8y = 1
Add 7 to both sides.
8y = 8
Divide both sides by 8.
y = 1
The solution is (-1, 1).
6. Answer :
2x - 3y = 3 ----A
-2x + 3y = 1 ----B
Add A and B.
A + B :
(2x - 3y) + (-2x + 3y) = 3 + 1
2x - 3y - 2x + 3y = 4
0 = 4 (false)
In the above step of solving the given system, there is no variable and '0 = 4' is false.
So, the given system of linear equations has NO solution.
7. Answer :
x - 2y = -5 ----A
-7x + 14y = 35 ----B
In B, the coefficients of x, y and the costant term are all evenly divisible by 7.
So, divide both sides of B by 7.
⁽⁻⁷ˣ ⁺ ¹⁴ʸ⁾⁄₇ = ³⁵⁄₇
⁽⁻⁷ˣ⁾⁄₇ ⁺ ⁽¹⁴ʸ⁾⁄₇ = 5
-x + 2y = 5 ----C
Add A and C.
A + C :
(x - 2y) + (-x + 2y) = -5 + 5
x - 2y - x + 2y = 0
0 = 0 (true)
In the above step of solving the given system, there is no variable and '0 = 0' is true.
So, the given system of linear equations has infinitely many solutions.
8. Answer :
Let x and y be the two numbers
Given : Sum of the two numbers is 18 and the difference between them is 2.
x + y = 18 ----A
x - y = 2 ----B
Add A and B to eliminate y.
(x + y) + (x - y) = 18 + 2
x + y + x - y = 20
2x = 20
Divide both sides by 2.
x = 10
Substitute x = 10 into A.
10 + y = 18
Subtract 10 from both sides.
y = 8
The numbers are 8 and 10.
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