Worksheet Solve systems of equations with fractions by Mrs Coverts ... - Free Printable

Let’s solve each system of equations step by step. We’ll use substitution or elimination — whichever is easier for each pair.

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Problem 1:

Equation A:
(2/5)x + (1/5)y = 5

Equation B:
x - y = 6 → so x = y + 6

Substitute x = y + 6 into Equation A:

(2/5)(y + 6) + (1/5)y = 5
→ (2/5)y + 12/5 + (1/5)y = 5
→ (3/5)y + 12/5 = 5
Multiply everything by 5 to eliminate denominators:
3y + 12 = 25
3y = 13
y = 13/3

Now plug back into x = y + 6:
x = 13/3 + 6 = 13/3 + 18/3 = 31/3

✔ Check:
(2/5)(31/3) + (1/5)(13/3) = (62/15) + (13/15) = 75/15 = 5 ✔️
x - y = 31/3 - 13/3 = 18/3 = 6 ✔️

Answer for #1: x = 31/3, y = 13/3

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Problem 2:

Equation A:
(1/5)x + (1/4)y = 5

Equation B:
(1/10)x + (3/5)y = 7

Let’s eliminate fractions first. Multiply Equation A by 20 (LCM of 5 and 4):
20*(1/5)x + 20*(1/4)y = 20*5 → 4x + 5y = 100 → Call this A1

Multiply Equation B by 10 (LCM of 10 and 5):
10*(1/10)x + 10*(3/5)y = 10*7 → x + 6y = 70 → Call this B1

Now we have:
A1: 4x + 5y = 100
B1: x + 6y = 70 → solve for x: x = 70 - 6y

Plug into A1:
4(70 - 6y) + 5y = 100
280 - 24y + 5y = 100
280 - 19y = 100
-19y = -180
y = 180/19

Then x = 70 - 6*(180/19) = 70 - 1080/19 = (1330 - 1080)/19 = 250/19

✔ Check in original Equation A:
(1/5)(250/19) + (1/4)(180/19) = 50/19 + 45/19 = 95/19 = 5 ✔️
Equation B: (1/10)(250/19) + (3/5)(180/19) = 25/19 + 108/19 = 133/19 = 7 ✔️

Answer for #2: x = 250/19, y = 180/19

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Problem 3:

Equation A:
(3/5)x + (2/5)y = 10

Equation B:
(1/5)x + (1/5)y = 1

Multiply both equations by 5 to eliminate denominators:

A: 3x + 2y = 50
B: x + y = 5 → so x = 5 - y

Plug into A:
3(5 - y) + 2y = 50
15 - 3y + 2y = 50
15 - y = 50
-y = 35 → y = -35

Then x = 5 - (-35) = 40

✔ Check:
(3/5)(40) + (2/5)(-35) = 24 - 14 = 10 ✔️
(1/5)(40) + (1/5)(-35) = 8 - 7 = 1 ✔️

Answer for #3: x = 40, y = -35

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Problem 4:

Equation A:
(2/3)x + (1/3)y = 27

Equation B:
(1/3)x + (2/3)y = 7

Multiply both equations by 3:

A: 2x + y = 81
B: x + 2y = 21

Solve B for x: x = 21 - 2y

Plug into A:
2(21 - 2y) + y = 81
42 - 4y + y = 81
42 - 3y = 81
-3y = 39 → y = -13

Then x = 21 - 2(-13) = 21 + 26 = 47

✔ Check:
(2/3)(47) + (1/3)(-13) = 94/3 - 13/3 = 81/3 = 27 ✔️
(1/3)(47) + (2/3)(-13) = 47/3 - 26/3 = 21/3 = 7 ✔️

Answer for #4: x = 47, y = -13

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Final Answer:
1. x = 31/3, y = 13/3
2. x = 250/19, y = 180/19
3. x = 40, y = -35
4. x = 47, y = -13