Solving Simultaneous Equations by Substitution (examples ... - Free Printable

Let's carefully go through the problem and identify the error in the provided solution, then correct it step by step.

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✔ Given System of Equations:

$$
\begin{align*}
(1)\quad & x + 3y = 6 \\
(2)\quad & 2x + 8y = -12
\end{align*}
$$

We are solving this system using substitution, as shown in the image.

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Step-by-Step Solution (Corrected)

#### Step 1: Isolate a variable
From equation (1):
$$
x + 3y = 6
\Rightarrow x = 6 - 3y \quad \text{(Correct)}
$$

✔ This step is correct.

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#### Step 2: Substitute into the other equation
Substitute $ x = 6 - 3y $ into equation (2):
$$
2x + 8y = -12
\Rightarrow 2(6 - 3y) + 8y = -12
$$

Now simplify:
$$
12 - 6y + 8y = -12
\Rightarrow 12 + 2y = -12
$$

✔ This step is also correct so far.

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#### Step 3: Solve for $ y $
$$
12 + 2y = -12
\Rightarrow 2y = -12 - 12 = -24
\Rightarrow y = -12
$$

So far, so good — but now let’s check Step 4.

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#### Step 4: Substitute back to find $ x $
Use $ x = 6 - 3y $, with $ y = -12 $:
$$
x = 6 - 3(-12) = 6 + 36 = 42
$$

So $ x = 42 $

This matches the given solution.

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#### Step 5: Check the solution

Plug $ x = 42 $, $ y = -12 $ into both equations.

Check equation (1):
$$
x + 3y = 42 + 3(-12) = 42 - 36 = 6 \quad \text{✓ Correct}
$$

Check equation (2):
$$
2x + 8y = 2(42) + 8(-12) = 84 - 96 = -12 \quad \text{✓ Correct}
$$

✔ So both equations are satisfied.

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✘ But Wait! There's an Issue...

The problem is not in the math — the math is actually correct. However, the solution seems unreasonable because:

- The values $ x = 42 $, $ y = -12 $ seem large.
- But they satisfy both equations, so they are correct.

But let’s double-check the substitution step again.

Wait — no, the algebra is correct.

Let’s verify once more:

Recheck Step 3:

$$
2(6 - 3y) + 8y = -12 \\
12 - 6y + 8y = -12 \\
12 + 2y = -12 \\
2y = -24 \\
y = -12 \quad \text{✓}
$$

Then:
$$
x = 6 - 3(-12) = 6 + 36 = 42 \quad \text{✓}
$$

And check:
- $ x + 3y = 42 + 3(-12) = 42 - 36 = 6 $ ✓
- $ 2x + 8y = 84 - 96 = -12 $ ✓

✔ So the solution is correct.

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🚨 But Why Does It Feel Off?

Because the numbers are large, but that doesn’t make them wrong.

Let’s see if we can solve it another way to confirm.

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Alternate Method: Elimination (for verification)

Given:
$$
\begin{align*}
(1)\quad & x + 3y = 6 \\
(2)\quad & 2x + 8y = -12
\end{align*}
$$

Multiply equation (1) by 2:
$$
2x + 6y = 12
$$

Now subtract from equation (2):
$$
(2x + 8y) - (2x + 6y) = -12 - 12 \\
2y = -24 \Rightarrow y = -12
$$

Then $ x = 6 - 3(-12) = 6 + 36 = 42 $

✔ Same result.

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✔ Final Answer:

$$
\boxed{x = 42,\quad y = -12}
$$

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🔍 Conclusion:

The solution shown in the image is actually correct, despite the large numbers.

However, there is a typo or formatting issue in Step 5:
It says:
> $ 42 = 6 - 3(-12) $

But that’s not how you check. You should plug into the original equation:

Correct check:
$$
x + 3y = 42 + 3(-12) = 42 - 36 = 6 \quad \text{✓}
$$

So instead of writing:
> $ 42 = 6 - 3(-12) $

You should write:
> $ x + 3y = 42 + 3(-12) = 6 $, which matches the first equation.

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✔ Corrected Step 5:

> Check:
> $ x + 3y = 42 + 3(-12) = 42 - 36 = 6 $ → True
> $ 2x + 8y = 2(42) + 8(-12) = 84 - 96 = -12 $ → True

✔️ Solution verified.

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✔ Final Answer:

$$
\boxed{x = 42,\quad y = -12}
$$

The method is correct, the steps are correct, and the solution checks out. The only issue is a miswritten check in Step 5.