Worksheet featuring six systems of linear equations to solve.

Let's solve each system of equations step by step. We'll use substitution or elimination methods depending on which is more efficient.

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1.

Given:
- $ y = 5x - 11 $
- $ y = -2x + 10 $

Since both expressions equal $ y $, set them equal to each other:

$$
5x - 11 = -2x + 10
$$

Add $ 2x $ to both sides:

$$
7x - 11 = 10
$$

Add 11 to both sides:

$$
7x = 21
$$

Divide by 7:

$$
x = 3
$$

Now substitute $ x = 3 $ into one of the original equations (e.g., $ y = 5x - 11 $):

$$
y = 5(3) - 11 = 15 - 11 = 4
$$

✔ Solution: $ (3, 4) $

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2.

Given:
- $ 7x - 3y = -24 $
- $ 3x - 7y = -16 $

We'll use elimination. Let's eliminate one variable.

Multiply the first equation by 7 and the second by 3:

- $ 7(7x - 3y) = 7(-24) \Rightarrow 49x - 21y = -168 $
- $ 3(3x - 7y) = 3(-16) \Rightarrow 9x - 21y = -48 $

Now subtract the second from the first:

$$
(49x - 21y) - (9x - 21y) = -168 - (-48)
$$
$$
49x - 21y - 9x + 21y = -168 + 48
$$
$$
40x = -120
\Rightarrow x = -3
$$

Substitute $ x = -3 $ into one of the original equations, say $ 7x - 3y = -24 $:

$$
7(-3) - 3y = -24 \Rightarrow -21 - 3y = -24
\Rightarrow -3y = -3 \Rightarrow y = 1
$$

✔ Solution: $ (-3, 1) $

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3.

Given:
- $ y = -3x + 6 $
- $ 2x - 3y = 4 $

Substitute $ y $ from the first equation into the second:

$$
2x - 3(-3x + 6) = 4
\Rightarrow 2x + 9x - 18 = 4
\Rightarrow 11x - 18 = 4
\Rightarrow 11x = 22
\Rightarrow x = 2
$$

Now find $ y $:

$$
y = -3(2) + 6 = -6 + 6 = 0
$$

✔ Solution: $ (2, 0) $

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4.

Given:
- $ y = x - 4 $
- $ x + 2y = 1 $

Substitute $ y = x - 4 $ into the second equation:

$$
x + 2(x - 4) = 1
\Rightarrow x + 2x - 8 = 1
\Rightarrow 3x - 8 = 1
\Rightarrow 3x = 9
\Rightarrow x = 3
$$

Now find $ y $:

$$
y = 3 - 4 = -1
$$

✔ Solution: $ (3, -1) $

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5.

Given:
- $ 4x - 5y = -10 $
- $ 4x - 5y = -5 $

Wait! Both equations are:

- $ 4x - 5y = -10 $
- $ 4x - 5y = -5 $

But this means:

$$
4x - 5y = -10 \quad \text{and} \quad 4x - 5y = -5
$$

This implies $ -10 = -5 $, which is false.

So there is no solution — the lines are parallel and never intersect.

✘ Solution: No solution (inconsistent system)

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6.

Given:
- $ 3y = 4x - 3 $ → rewrite as $ -4x + 3y = -3 $
- $ 8x - 6y = 6 $

Let’s simplify and use elimination.

First equation:
$ -4x + 3y = -3 $

Second equation:
$ 8x - 6y = 6 $

Notice that if we multiply the first equation by 2:

$$
2(-4x + 3y) = 2(-3) \Rightarrow -8x + 6y = -6
$$

Now add this to the second equation:

$$
(-8x + 6y) + (8x - 6y) = -6 + 6
\Rightarrow 0 = 0
$$

This is always true — meaning the two equations represent the same line (dependent system).

So there are infinitely many solutions.

Let’s write in terms of one variable.

From $ 3y = 4x - 3 $, solve for $ y $:

$$
y = \frac{4x - 3}{3}
$$

So all points $ \left(x, \frac{4x - 3}{3}\right) $ satisfy the system.

✔ Solution: Infinitely many solutions, given by $ y = \frac{4x - 3}{3} $

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✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| 1 | $ (3, 4) $ |
| 2 | $ (-3, 1) $ |
| 3 | $ (2, 0) $ |
| 4 | $ (3, -1) $ |
| 5 | No solution |
| 6 | Infinitely many solutions: $ y = \frac{4x - 3}{3} $ |

Let me know if you'd like a graph or explanation of any particular one!