Standard algebra worksheet designed to help students practice the elimination method for solving linear systems.

You’ve uploaded a worksheet titled “Solving Systems of Equations by Elimination Method”. There are 6 systems of equations to solve using elimination. I’ll solve each one step-by-step and explain the method.

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## 🔁 Elimination Method Overview:
The goal is to eliminate one variable by adding or subtracting the equations (after possibly multiplying one or both equations by constants), then solve for the remaining variable. Finally, substitute back to find the other variable.

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✔ Problem 1:

> x + 3y = 0
> –x + 2y = 5

Step 1: Add the equations to eliminate x.
Notice that the coefficients of `x` are +1 and –1 — they cancel when added.

```
x + 3y = 0
+ (-x + 2y = 5)
---------------
5y = 5
```

Step 2: Solve for y.
→ `y = 1`

Step 3: Substitute y = 1 into first equation to find x.
`x + 3(1) = 0` → `x + 3 = 0` → `x = -3`

✔ Solution: (x, y) = (-3, 1)

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✔ Problem 2:

> x + 3y = 0
> x + 4y = 1

Step 1: Subtract the first equation from the second to eliminate x.

```
x + 4y = 1
- (x + 3y = 0)
---------------
y = 1
```

Step 2: Plug y = 1 into first equation.
`x + 3(1) = 0` → `x = -3`

✔ Solution: (x, y) = (-3, 1)

*(Same as problem 1 — different system, same solution!)*

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✔ Problem 3:

> 6x + 2y = 8
> 12x + 2y = 8

Step 1: Subtract the first equation from the second to eliminate y.

```
12x + 2y = 8
- (6x + 2y = 8)
----------------
6x = 0
```

→ `x = 0`

Step 2: Plug x = 0 into first equation.
`6(0) + 2y = 8` → `2y = 8` → `y = 4`

✔ Solution: (x, y) = (0, 4)

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✔ Problem 4:

> 3x – 5y = 0
> –4x – 5y = 0

Step 1: Subtract the second equation from the first to eliminate y.

```
3x – 5y = 0
- (–4x – 5y = 0)
-----------------
7x = 0
```

→ `x = 0`

Step 2: Plug x = 0 into first equation.
`3(0) – 5y = 0` → `–5y = 0` → `y = 0`

✔ Solution: (x, y) = (0, 0)

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✔ Problem 5:

> 3x + 2y = 0
> 5x – 2y = 0

Step 1: Add the equations to eliminate y.

```
3x + 2y = 0
+ 5x – 2y = 0
---------------
8x = 0
```

→ `x = 0`

Step 2: Plug x = 0 into first equation.
`3(0) + 2y = 0` → `2y = 0` → `y = 0`

✔ Solution: (x, y) = (0, 0)

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✔ Problem 6:

> –5x – 4y = 0
> –5x + 4y = 2

Step 1: Add the equations to eliminate y.

```
–5x – 4y = 0
+ –5x + 4y = 2
----------------
–10x = 2
```

→ `x = –2/10 = –1/5`

Step 2: Plug x = –1/5 into first equation.
`–5(–1/5) – 4y = 0` → `1 – 4y = 0` → `–4y = –1` → `y = 1/4`

✔ Solution: (x, y) = (–1/5, 1/4)

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## 📝 Final Answers:

1. (–3, 1)
2. (–3, 1)
3. (0, 4)
4. (0, 0)
5. (0, 0)
6. (–1/5, 1/4)

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## 💡 Tips for Elimination Method:
- Look for variables with opposite coefficients (add to eliminate) or same coefficients (subtract to eliminate).
- If needed, multiply one or both equations by a constant to make coefficients match.
- Always check your solution by plugging it back into both original equations!

Let me know if you’d like to see these solved graphically or by substitution too!