I'll solve each problem step by step.
Problem 1:
x + 2y = 12
3x + 4y = 4
From equation 1: x = 12 - 2y
Substitute into equation 2:
3(12 - 2y) + 4y = 4
36 - 6y + 4y = 4
36 - 2y = 4
-2y = -32
y = 16
x = 12 - 2(16) = 12 - 32 = -20
Answer: x = -20, y = 16
Problem 2:
x + y = 10
x - y = 20
Add the equations:
2x = 30
x = 15
Substitute: 15 + y = 10
y = -5
Answer: x = 15, y = -5
Problem 3:
2x + y = 15
x - y = 45
Add the equations:
3x = 60
x = 20
Substitute: 20 - y = 45
-y = 25
y = -25
Answer: x = 20, y = -25
Problem 4:
4x + 3y = 32
2x - 7y = -18
Multiply second equation by 2:
4x - 14y = -36
Subtract from first equation:
(4x + 3y) - (4x - 14y) = 32 - (-36)
17y = 68
y = 4
Substitute: 4x + 3(4) = 32
4x + 12 = 32
4x = 20
x = 5
Answer: x = 5, y = 4
Problem 5:
0.4x - 0.7y = -13
0.6x + 0.7y = 33
Add the equations:
x = 20
Substitute: 0.4(20) - 0.7y = -13
8 - 0.7y = -13
-0.7y = -21
y = 30
Answer: x = 20, y = 30
Problem 6:
2x + 5y = 6xy ... (1)
4x - 5y = -3xy ... (2)
Add equations:
6x = 3xy
Divide by 3x (assuming x ≠ 0):
2 = y
Substitute y = 2 into equation 1:
2x + 5(2) = 6x(2)
2x + 10 = 12x
10 = 10x
x = 1
Sum: x + y = 1 + 2 = 3
Answer: 3
Problem 7:
Let the two-digit number be 10a + b, where a and b are digits.
Given: |a - b| = 3
Reversed number: 10b + a
4(10b + a) = 7(10a + b)
40b + 4a = 70a + 7b
33b = 66a
b = 2a
Since |a - b| = 3 and b = 2a:
If b > a: 2a - a = 3, so a = 3, b = 6
Number = 36
Check: Reversed = 63
4 × 63 = 252
7 × 36 = 252 ✓
Answer: 36
Problem 8:
Container 1: 24L milk, 6L water (ratio 4:1)
Container 2: 15L milk, 10L water (ratio 3:2)
Container 3: 25L milk, 10L water
Let x liters from container 1, y liters from container 2.
Milk in x liters from C1: (24/30)x = 0.8x
Water in x liters from C1: (6/30)x = 0.2x
Milk in y liters from C2: (15/25)y = 0.6y
Water in y liters from C2: (10/25)y = 0.4y
Total milk: 0.8x + 0.6y = 25
Total water: 0.2x + 0.4y = 10
Multiply second equation by 2:
0.4x + 0.8y = 20
Multiply first equation by 2:
1.6x + 1.2y = 50
From second original: 0.2x + 0.4y = 10
Multiply by 4: 0.8x + 1.6y = 40
Subtract from first:
(1.6x + 1.2y) - (0.8x + 1.6y) = 50 - 40
0.8x - 0.4y = 10
From 0.2x + 0.4y = 10:
0.4y = 10 - 0.2x
y = 25 - 0.5x
Substitute: 0.8x + 0.6(25 - 0.5x) = 25
0.8x + 15 - 0.3x = 25
0.5x = 10
x = 20
Answer: 20 liters
Final Answer:
1) x = -20, y = 16
2) x = 15, y = -5
3) x = 20, y = -25
4) x = 5, y = 4
5) x = 20, y = 30
6) 3
7) 36
8) 20
Parent Tip: Review the logic above to help your child master the concept of equations in two variables worksheet answers.
