Problem 396 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{4 x-3}=7\)

Step-by-Step Solution

Answer

x = 13

Step 1: Isolate the square root

The equation \(\text{\textbackslash sqrt\textbraceleft 4x - 3\textbraceright = 7\) already has the square root isolated.

Step 2: Square Both Sides

Eliminate the square root by squaring both sides of the equation. This gives \[ (\text{\textbackslash sqrt\textbraceleft 4x - 3\textbraceright})^2 = 7^2 \Rightarrow 4x - 3 = 49 \]

Step 3: Solve for x

Add 3 to both sides of the equation to isolate the term with x: \[ 4x - 3 + 3 = 49 + 3 \Rightarrow 4x = 52 \]

Step 4: Divide by the Coefficient of x

Divide both sides by 4 to solve for x: \[ x = \frac{52}{4} = 13 \]

Key Concepts

Square Root

A square root in mathematics helps find a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 * 5 = 25. In our exercise, \(\sqrt{4x-3} = 7\), the expression under the square root of 4x-3 must equal 49 because \(\sqrt{49} = 7\). Square roots help in balancing equations and simplifying expressions.

Algebraic Isolation

Algebraic isolation involves rearranging the equation to get the variable by itself on one side. In our exercise, the square root expression \(\sqrt{4x-3}\) is already isolated. When isolating terms, perform inverse operations, like adding, subtracting, or dividing, to clear out other terms. This helps focus on solving for the variable.

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. We start by isolating terms, as we did with \(4x-3\) after squaring both sides. Solving involves performing operations on both sides of the equation to maintain equality. In the exercise: \(4x-3=49\), we added 3 to both sides to isolate the term with the variable \(4x\), giving \(4x=52\). Finally, we divided by 4 to find \(x=13\).

Squaring Both Sides

Squaring both sides of an equation removes the square root. For instance, with \(\sqrt{4x-3}=7\), squaring transforms it to \((4x-3)=7^2\). This operation is crucial for eliminating square roots and simplifying the equation to a more manageable form. Ensure to square both sides equally to maintain the equation's balance.