Solving Systems of Equations by Substitution Worksheets - Math Monks - Free Printable

Explanation:
We are solving systems of equations using substitution. That means we replace one variable in an equation with an expression from the other equation.

Let’s solve each system one by one, carefully.

---

1)
x = –2y
x – y = 9

Substitute x = –2y into the second equation:
(–2y) – y = 9
–3y = 9
y = –3

Now plug y = –3 back into x = –2y:
x = –2(–3) = 6

✔ Solution: (x, y) = (6, –3)

---

2)
y = x – 4
–4x – 6y = –16

Substitute y = x – 4 into the second equation:
–4x – 6(x – 4) = –16
–4x – 6x + 24 = –16
–10x + 24 = –16
–10x = –40
x = 4

Then y = x – 4 = 4 – 4 = 0

✔ Solution: (4, 0)

---

3)
–x – 3y = –18
y = –4x + 6

Substitute y = –4x + 6 into first equation:
–x – 3(–4x + 6) = –18
–x + 12x – 18 = –18
11x – 18 = –18
11x = 0
x = 0

Then y = –4(0) + 6 = 6

✔ Solution: (0, 6)

---

4)
–2x + 2y = 6
4x + 2y = –5

Let’s solve one for y or x. Let’s isolate y from first equation:
–2x + 2y = 6 → divide both sides by 2:
–x + y = 3 → y = x + 3

Now substitute into second equation:
4x + 2(x + 3) = –5
4x + 2x + 6 = –5
6x + 6 = –5
6x = –11
x = –11/6

Then y = x + 3 = –11/6 + 18/6 = 7/6

✔ Solution: (–11/6, 7/6)

Wait — let’s double-check this because fractions are tricky.

Original equations:
1) –2x + 2y = 6
2) 4x + 2y = –5

Subtract equation (1) from equation (2):
(4x + 2y) – (–2x + 2y) = –5 – 6
4x + 2y + 2x – 2y = –11
6x = –11 → x = –11/6 ✔️
Then plug into eq (1):
–2(–11/6) + 2y = 6
22/6 + 2y = 6
11/3 + 2y = 6
2y = 6 – 11/3 = 18/3 – 11/3 = 7/3
y = 7/6 ✔️

So correct.

---

5)
y = –6
3x – 6y = 30

Substitute y = –6:
3x – 6(–6) = 30
3x + 36 = 30
3x = –6
x = –2

✔ Solution: (–2, –6)

---

6)
–2x – y = –5
x – 8y = –23

Solve first for y:
–2x – y = –5 → multiply both sides by –1:
2x + y = 5 → y = 5 – 2x

Substitute into second equation:
x – 8(5 – 2x) = –23
x – 40 + 16x = –23
17x – 40 = –23
17x = 17
x = 1

Then y = 5 – 2(1) = 3

✔ Solution: (1, 3)

Double-check:
–2(1) – 3 = –2 – 3 = –5 ✔️
1 – 8(3) = 1 – 24 = –23 ✔️

---

7)
2x + y = 2
3x + 7y = 14

Solve first for y:
y = 2 – 2x

Substitute into second:
3x + 7(2 – 2x) = 14
3x + 14 – 14x = 14
–11x + 14 = 14
–11x = 0
x = 0

Then y = 2 – 2(0) = 2

✔ Solution: (0, 2)

Check:
2(0) + 2 = 2 ✔️
3(0) + 7(2) = 14 ✔️

---

8)
y = 4x – 9
y = x – 3

Since both equal y, set them equal:
4x – 9 = x – 3
4x – x = –3 + 9
3x = 6
x = 2

Then y = x – 3 = 2 – 3 = –1

✔ Solution: (2, –1)

Check:
y = 4(2) – 9 = 8 – 9 = –1 ✔️
y = 2 – 3 = –1 ✔️

---

All solutions verified.

Final Answer:
1) (6, –3)
2) (4, 0)
3) (0, 6)
4) (–11/6, 7/6)
5) (–2, –6)
6) (1, 3)
7) (0, 2)
8) (2, –1)