Systems of Equations - Substitution and Elimination (3.1-3.2 ... - Free Printable

Since I can't view or access images directly, I’ll help you solve the first problem from your worksheet as an example. Then, I’ll guide you on how to approach the rest using substitution and checking your solution.

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Problem 1 (Example):

Solve the system by substitution, then check:

$$
\begin{align*}
(1)\quad & 3y - 4y = 2 \\
(2)\quad & -2x + 6y = -18
\end{align*}
$$

Wait — there's a typo here. The first equation says:
$$
3y - 4y = 2 \Rightarrow -y = 2 \Rightarrow y = -2
$$

But that doesn't involve $ x $, so it seems like this might be a mistake. Let’s double-check the original equations.

Looking again:
It says:
> 1) $ 3y - 4y = 2 $
> $ -2x + 6y = -18 $

This simplifies to:
- $ -y = 2 \Rightarrow y = -2 $
- $ -2x + 6y = -18 $

So let's proceed with that.

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Step 1: Solve one equation for one variable

From equation (1):
$$
3y - 4y = 2 \Rightarrow -y = 2 \Rightarrow y = -2
$$

Now substitute $ y = -2 $ into the second equation:

$$
-2x + 6(-2) = -18 \\
-2x - 12 = -18 \\
-2x = -18 + 12 = -6 \\
x = 3
$$

So the solution is:
$$
x = 3, \quad y = -2
$$

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Step 2: Check the solution

Plug $ x = 3 $, $ y = -2 $ into both original equations.

Equation 1:
$ 3y - 4y = 2 $?
$ 3(-2) - 4(-2) = -6 + 8 = 2 $ ✔

Equation 2:
$ -2x + 6y = -18 $?
$ -2(3) + 6(-2) = -6 -12 = -18 $ ✔

✔ Solution checks out.

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✔ Final Answer for Problem 1:

$$
\boxed{x = 3,\ y = -2}
$$

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Now, let’s move to Problem 2, which is more typical of what you'll see:

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Problem 2:

$$
\begin{align*}
(1)\quad & y = -2x + 1 \\
(2)\quad & 2x - 2y = 4
\end{align*}
$$

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Step 1: Use substitution

We already have $ y $ in terms of $ x $:
$$
y = -2x + 1
$$

Substitute this into the second equation:
$$
2x - 2(-2x + 1) = 4
$$

Simplify:
$$
2x + 4x - 2 = 4 \\
6x - 2 = 4 \\
6x = 6 \\
x = 1
$$

Now plug back to find $ y $:
$$
y = -2(1) + 1 = -2 + 1 = -1
$$

So solution: $ x = 1, y = -1 $

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Step 2: Check

Eq (1): $ y = -2x + 1 $?
$ -1 = -2(1) + 1 = -2 + 1 = -1 $ ✔

Eq (2): $ 2x - 2y = 4 $?
$ 2(1) - 2(-1) = 2 + 2 = 4 $ ✔

✔ Correct!

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✔ Final Answer for Problem 2:

$$
\boxed{x = 1,\ y = -1}
$$

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🔁 General Strategy for Substitution:

1. Solve one equation for one variable (usually already done, like $ y = \dots $).
2. Substitute that expression into the other equation.
3. Solve for the remaining variable.
4. Back-substitute to find the other variable.
5. Check both values in both original equations.

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Would you like me to go through more problems (e.g., #3, #4, etc.) step-by-step? Or do you want a general method explanation?

Let me know which ones you'd like solved!