Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 4=7-\sqrt{33 x-2} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{1}{3}\).

Step 1: Isolate the square root term

To isolate the square root term, subtract 4 from both sides of the equation: \(7 - 4 = \sqrt{33x-2}\), which simplifies to: \(3 = \sqrt{33x-2}\).

Step 2: Eliminate the square root

Square both sides of the equation to get rid of the square root. This gives us: \(3^2 = (\sqrt{33x-2})^2\), which simplifies to: \(9 = 33x - 2\).

Step 3: Solve for x

Add 2 to both sides of the equation: \(33x = 9 + 2\), which gives us: \(33x = 11\). Now divide both sides of the equation by 33 to solve for 'x'. Thus, \(x = \frac{11}{33}\).

Key Concepts

Isolating Square Roots

Isolating a square root in an equation is often the first step to solving it. The objective is to have the square root term by itself on one side of the equation. This makes the equation manageable so you can later eliminate the square root. Let's look at how it's done.

• You start by identifying which part of the equation contains the square root. In our example, the term \( \sqrt{33x - 2} \) needs to be isolated.
• Once identified, carefully arrange your equation so the square root is on its own by performing operations like addition, subtraction, multiplication, or division.
• For the exercise we're working on, we subtracted 7 from both sides to achieve isolation of the square root: \( 7 - 4 = \sqrt{33x-2} \).

With the square root isolated, you're set to proceed to the next step, which is squaring both sides.

Squaring Both Sides

Once you've successfully isolated the square root, the next move is to square both sides of the equation. This will remove the square root, allowing you to turn your focus on solving for the variable.

• When you square the isolated square root term, the square root vanishes. Mathematically, for \( \sqrt{a} \), squaring both sides would give \( a \).
• For our example, squaring both sides results in \( 3^2 = (\sqrt{33x-2})^2 \), leading to \( 9 = 33x - 2 \).
• With the square root gone, you now have a simpler equation that you can work with.

This process forms the transition from dealing with square roots to solving linear equations, making it a critical step.

Linear Equations

Understanding linear equations is crucial as they appear often in various math problems. A linear equation is an equation between two variables that gives a straight line when plotted on a graph.

• Once the square root is eliminated, you often end up with a linear equation, as we saw: \( 9 = 33x - 2 \).
• The next task is simplifying the equation to solve for the unknown variable. This usually involves rearranging the equation by performing operations such as addition, subtraction, multiplication, and division.
• In our case, after adding 2 to both sides to get \( 33x = 11 \), we divided both sides by 33 to solve for \( x \), achieving \( x = \frac{11}{33} \).
• Remember, the goal is to isolate the variable, leaving you with its solution.

Always double-check your solution by plugging it back into the original equation. This ensures your answer satisfies the given conditions.