Solving Equations Worksheets - Free Printable

Let's solve one of these simultaneous equations using the elimination method, and then I’ll explain the general approach so you can apply it to the rest.

We'll go through Problem 1 step by step:

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🔷 Problem 1:

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

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✔ Step 1: Make the coefficients of one variable the same

We want to eliminate one variable. Let’s eliminate $ x $. Both equations have $ x $ with coefficient 1, so we can subtract one equation from the other.

But let’s align them:

Equation (1): $ x + 3y = 6 $
Equation (2): $ x - y = 2 $

Now subtract Equation (2) from Equation (1):

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

Simplify:

$$
x + 3y - x + y = 4 \Rightarrow 4y = 4
$$

So:
$$
y = 1
$$

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✔ Step 2: Substitute back to find $ x $

Use Equation (2): $ x - y = 2 $

Substitute $ y = 1 $:

$$
x - 1 = 2 \Rightarrow x = 3
$$

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✔ Final Answer for Problem 1:

$$
\boxed{x = 3,\quad y = 1}
$$

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📌 General Elimination Method Summary:

To solve a pair of simultaneous equations using elimination:

1. Choose a variable to eliminate (either $ x $ or $ y $).
2. Make the coefficients of that variable the same in both equations by multiplying each equation if needed.
3. Add or subtract the equations to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute back into one of the original equations to find the other variable.
6. Check your solution in both equations.

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Let’s do Problem 2 quickly as another example:

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🔷 Problem 2:

$$
\begin{align*}
(1)\quad & 3x + 2y = 9 \\
(2)\quad & x - 2y = -5
\end{align*}
$$

We notice that $ 2y $ and $ -2y $ are already opposites. So add the equations:

$$
(3x + 2y) + (x - 2y) = 9 + (-5)
\Rightarrow 4x = 4 \Rightarrow x = 1
$$

Now substitute $ x = 1 $ into (2):

$$
1 - 2y = -5 \Rightarrow -2y = -6 \Rightarrow y = 3
$$

✔ Check: Plug into (1): $ 3(1) + 2(3) = 3 + 6 = 9 $ → Correct!

Answer: $ \boxed{x = 1,\ y = 3} $

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Would you like me to solve all 18 problems step-by-step? Or would you prefer a few more examples and then you try the rest?

Let me know how you'd like to proceed!