Let's go through solving the equation [tex]\(\sqrt{6y} = \sqrt{36 + 2y}\)[/tex] step-by-step, matching each step with its justification.
### Step 1: Squaring Property of Equality
Given:
[tex]\[ \sqrt{6y} = \sqrt{36 + 2y} \][/tex]
To eliminate the square roots, we square both sides of the equation.
[tex]\[ (\sqrt{6y})^2 = (\sqrt{36 + 2y})^2 \][/tex]
This simplifies to:
[tex]\[ 6y = 36 + 2y \][/tex]
This step is justified by the squaring property of equality.
### Step 2: Subtraction Property of Equality
Next, we want to isolate the variable [tex]\(y\)[/tex]. We do this by subtracting [tex]\(2y\)[/tex] from both sides of the equation.
[tex]\[ 6y - 2y = 36 \][/tex]
This simplifies to:
[tex]\[ 4y = 36 \][/tex]
This step is justified by the subtraction property of equality.
### Step 3: Division Property of Equality
Finally, to solve for [tex]\(y\)[/tex], we divide both sides of the equation by 4.
[tex]\[ y = \frac{36}{4} \][/tex]
This simplifies to:
[tex]\[ y = 9 \][/tex]
This step is justified by the division property of equality.
### Summary
- Squaring Property of Equality: [tex]\( \sqrt{6y} = \sqrt{36 + 2y} \)[/tex] simplifies to [tex]\( 6y = 36 + 2y \)[/tex]
- Subtraction Property of Equality: [tex]\( 6y = 36 + 2y \)[/tex] simplifies to [tex]\( 4y = 36 \)[/tex]
- Division Property of Equality: [tex]\( 4y = 36 \)[/tex] simplifies to [tex]\( y = 9 \)[/tex]
So, the correct matching of steps with justifications is:
- [tex]\( 6y = 36 + 2y \)[/tex]: squaring property of equality
- [tex]\( 4y = 36 \)[/tex]: subtraction property of equality
- [tex]\( y = 9 \)[/tex]: division property of equality
Therefore, the final solution is: [tex]\( y = 9 \)[/tex].
