This math worksheet provides a step-by-step guide to the elimination method, including a solved example and practice problems for solving linear systems.

Let’s solve each system of equations using the elimination method — that means we’ll add or subtract the equations to cancel out one variable, then solve for the other.

---

Problem 1:

Equations:
2x + 2y = -2
3x - 2y = 12

Notice: The y terms are +2y and -2y → they are opposites! So if we add the two equations, the y’s will cancel.

Add them:

(2x + 3x) + (2y - 2y) = -2 + 12
→ 5x + 0 = 10
→ 5x = 10
→ x = 2

Now plug x = 2 into one of the original equations. Let’s use the first one:

2x + 2y = -2
2(2) + 2y = -2
4 + 2y = -2
2y = -6
y = -3

✔ Solution: (2, -3)

---

Problem 2:

Equations:
4x - 2y = -1
-4x + 4y = -2

Look at the x terms: +4x and -4x → perfect! Add the equations to eliminate x.

Add them:

(4x - 4x) + (-2y + 4y) = -1 + (-2)
→ 0 + 2y = -3
→ 2y = -3
→ y = -3/2 or -1.5

Now plug y = -3/2 into the first equation:

4x - 2(-3/2) = -1
4x + 3 = -1
4x = -4
x = -1

✔ Solution: (-1, -3/2) or (-1, -1.5)

*(We can write it as a fraction unless told otherwise.)*

---

Problem 3:

Equations:
x - y = 2
x + y = -3

The y terms: -y and +y → add to eliminate y.

Add them:

(x + x) + (-y + y) = 2 + (-3)
→ 2x + 0 = -1
→ 2x = -1
→ x = -1/2 or -0.5

Plug x = -1/2 into second equation:

x + y = -3
-1/2 + y = -3
y = -3 + 1/2 = -5/2 or -2.5

✔ Solution: (-1/2, -5/2) or (-0.5, -2.5)

---

Problem 4:

Equations:
6x + 5y = 4
6x - 7y = -20

Both have 6x, so if we subtract the second equation from the first, the x’s will cancel.

Subtract:

(6x - 6x) + (5y - (-7y)) = 4 - (-20)
→ 0 + (5y + 7y) = 4 + 20
→ 12y = 24
→ y = 2

Now plug y = 2 into first equation:

6x + 5(2) = 4
6x + 10 = 4
6x = -6
x = -1

✔ Solution: (-1, 2)

---

Problem 5:

Equations:
2x - 3y = 12
4x + 3y = 24

The y terms: -3y and +3y → add to eliminate y.

Add them:

(2x + 4x) + (-3y + 3y) = 12 + 24
→ 6x + 0 = 36
→ 6x = 36
→ x = 6

Plug x = 6 into first equation:

2(6) - 3y = 12
12 - 3y = 12
-3y = 0
y = 0

✔ Solution: (6, 0)

---

Final Answer:
1. (2, -3)
2. (-1, -3/2)
3. (-1/2, -5/2)
4. (-1, 2)
5. (6, 0)