Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 3-\sqrt{y}=1 $$

Step-by-Step Solution

Answer

The solution set for the equation is \(\{4\}\).

Step 1: Isolate the Square Root Expression

Start by isolating the square root on one side of the equation. We have: \[3 - \sqrt{y} = 1\]To isolate \(\sqrt{y}\), subtract 3 from both sides:\[-\sqrt{y} = 1 - 3\]Simplify the right-hand side:\[-\sqrt{y} = -2\]

Step 2: Eliminate the Negative Sign

Multiply both sides by -1 to eliminate the negative sign:\[\sqrt{y} = 2\]

Step 3: Solve for y

Remove the square root by squaring both sides of the equation:\[y = 2^2\]Calculate the square:\[y = 4\]

Step 4: Check the Solution

Substitute \(y = 4\) back into the original equation to verify the solution:\[3 - \sqrt{4} = 1\]Calculate \(\sqrt{4}\) which equals 2:\[3 - 2 = 1\]This simplifies to 1 = 1, which is true.

Step 5: Write the Solution Set

The solution satisfies the equation, so the solution set is:\[\{4\}\]

Key Concepts

Isolating the Variable

When solving equations, one of the first essential steps is isolating the variable. In this context, it means moving all other numbers or terms to the opposite side of the equation. This helps in clearly visualizing the problem you're solving for.

Let's start with the equation provided:

• Given: \[ 3 - \sqrt{y} = 1 \]
• First, we want to isolate the square root term. We do this by subtracting 3 from both sides of the equation: \[ -\sqrt{y} = 1 - 3 \]
• Simplify to: \[ -\sqrt{y} = -2 \]

This step ensures that the square root term stands alone on one side. Getting here might require basic arithmetic, but it's a crucial point in solving more complex equations.

Eliminating Square Roots

After successfully isolating the variable, it's time to eliminate the square root to find the variable's value.

The presence of a square root can prevent you from solving for the variable easily. To eliminate it:

• We have \[ -\sqrt{y} = -2 \]
• First, eliminate the negative sign by multiplying both sides by \[ -1 \]n \[ \sqrt{y} = 2 \]
• Square both sides of the equation to remove the square root:\[ y = (2)^2 \]
• This calculation shows: \[ y = 4 \]

By squaring both sides, the square root is nullified, making the variable accessible and easy to solve for.

Solution Set Verification

Once the variable is isolated and the square root is eliminated, the last step is solution verification. This ensures the obtained value satisfies the original equation, maintaining accuracy of your solution.

Here's how you can check:

• Substitute: Plug \[ y = 4 \] into the original equation:\[ 3 - \sqrt{4} = 1 \]
• Calculate \( \sqrt{4} \) to get \( 2 \)
• Subtraction: \[ 3 - 2 = 1 \] which simplifies to \[ 1 = 1 \], a true statement.
• Since the original equation holds true with \[ y = 4 \], we conclude that \( y = 4 \) is indeed a solution.
  

So, the solution set is: \[ \{ 4 \} \]. Solution set verification is essential to ensure that errors are avoided, providing confidence in your algebraic work.