Let’s solve each pair of linear equations step by step. We’ll use substitution or elimination — whichever is easier for each problem.
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Problem 1:
4x + 3y = 6
3x + 4y = 8
Let’s use elimination. Multiply first equation by 4 and second by 3 to make y coefficients match:
→ (4x + 3y = 6) × 4 → 16x + 12y = 24
→ (3x + 4y = 8) × 3 → 9x + 12y = 24
Now subtract the second from the first:
(16x + 12y) - (9x + 12y) = 24 - 24
7x = 0 → x = 0
Plug x = 0 into first equation:
4(0) + 3y = 6 → 3y = 6 → y = 2
✔ Check in second equation: 3(0) + 4(2) = 8 → 8 = 8 ✔️
Answer: x = 0, y = 2
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Problem 2:
6u + v = 18
5u + 2v = 22
Solve first equation for v:
v = 18 - 6u
Plug into second equation:
5u + 2(18 - 6u) = 22
5u + 36 - 12u = 22
-7u + 36 = 22
-7u = -14 → u = 2
Then v = 18 - 6(2) = 18 - 12 = 6
✔ Check: 5(2) + 2(6) = 10 + 12 = 22 ✔️
Answer: u = 2, v = 6
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Problem 3:
3p + 4q = 33
6p + 3q = 36
Multiply first equation by 2:
6p + 8q = 66
Subtract second equation:
(6p + 8q) - (6p + 3q) = 66 - 36
5q = 30 → q = 6
Plug into first equation:
3p + 4(6) = 33 → 3p + 24 = 33 → 3p = 9 → p = 3
✔ Check: 6(3) + 3(6) = 18 + 18 = 36 ✔️
Answer: p = 3, q = 6
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Problem 4:
2w + z = 13
w + z = 8
Subtract second equation from first:
(2w + z) - (w + z) = 13 - 8
w = 5
Plug into second: 5 + z = 8 → z = 3
✔ Check: 2(5) + 3 = 10 + 3 = 13 ✔️
Answer: w = 5, z = 3
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Problem 5:
2w + 3y = 12
2w + y = 6
Subtract second from first:
(2w + 3y) - (2w + y) = 12 - 6
2y = 6 → y = 3
Plug into second: 2w + 3 = 6 → 2w = 3 → w = 1.5
✔ Check: 2(1.5) + 3(3) = 3 + 9 = 12 ✔️
Answer: w = 1.5, y = 3
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Problem 6:
u + 6y = 32
u + 3y = 17
Subtract second from first:
(u + 6y) - (u + 3y) = 32 - 17
3y = 15 → y = 5
Plug into second: u + 3(5) = 17 → u + 15 = 17 → u = 2
✔ Check: 2 + 6(5) = 2 + 30 = 32 ✔️
Answer: u = 2, y = 5
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Problem 7:
c + 6d = 7
-c - 2d = -2
Add both equations together:
(c - c) + (6d - 2d) = 7 - 2
4d = 5 → d = 1.25
Plug into first: c + 6(1.25) = 7 → c + 7.5 = 7 → c = -0.5
✔ Check: -(-0.5) - 2(1.25) = 0.5 - 2.5 = -2 ✔️
Answer: c = -0.5, d = 1.25
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Problem 8:
8e + 7f = 43
2e - 7 = -f → Let’s rewrite this as: f = -2e + 7
Plug into first equation:
8e + 7(-2e + 7) = 43
8e -14e + 49 = 43
-6e + 49 = 43
-6e = -6 → e = 1
Then f = -2(1) + 7 = 5
✔ Check: 8(1) + 7(5) = 8 + 35 = 43 ✔️
Answer: e = 1, f = 5
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Final Answer:
1. x = 0, y = 2
2. u = 2, v = 6
3. p = 3, q = 6
4. w = 5, z = 3
5. w = 1.5, y = 3
6. u = 2, y = 5
7. c = -0.5, d = 1.25
8. e = 1, f = 5
Parent Tip: Review the logic above to help your child master the concept of 2 variable equations worksheet.
