Problem 128 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{3 x+4}=\sqrt{7 x+2} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Square Both Sides

To eliminate the square roots, square both sides of the equation: \(\left(\sqrt{3x + 4}\right)^2 = \left(\sqrt{7x + 2}\right)^2\)This transforms to:\(3x + 4 = 7x + 2\)

Step 2: Rearrange the Equation

Bring all the \(x\) terms to one side and the constant terms to the other side. Subtract \(3x\) from both sides:\(3x + 4 - 3x = 7x + 2 - 3x\)This simplifies to:\(4 = 4x + 2\)

Step 3: Solve for x

Subtract 2 from both sides to isolate the terms with \(x\):\(4 - 2 = 4x\)This results in:\(2 = 4x\)Divide both sides by 4 to solve for \(x\):\(x = \frac{2}{4} = \frac{1}{2}\)

Step 4: Verify the Solution

Substitute \(x = \frac{1}{2}\) back into the original equation to check for extraneous solutions. Calculate:\(\sqrt{3\left(\frac{1}{2}\right) + 4} = \sqrt{7\left(\frac{1}{2}\right) + 2}\)Simplify inside the square roots:\(\sqrt{1.5 + 4} = \sqrt{3.5 + 2}\)\(\sqrt{5.5} = \sqrt{5.5}\)Both sides are equal, so \(x = \frac{1}{2}\) is a valid solution.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value(s) of the variable(s) that satisfy the given equation. The goal is to manipulate the equation to isolate the variable on one side, ultimately finding its value. This usually involves a series of operations that maintain the equality of both sides of the equation.

• Identify the operations to undo: Consider the operations applied to the variable. These might include addition, subtraction, multiplication, division, or more complex operations like taking square roots.
• Perform inverse operations in reverse order: Start from the last operation applied and proceed backward. For example, if a variable was multiplied, divide to undo that operation.
• Maintain equality: Always perform the same operation on both sides to keep the equation balanced.

Practicing with different types of equations enhances critical thinking and problem-solving skills.

Square Root Equations

Square root equations involve expressions under a square root, and solving them often requires special techniques. The presence of the square root can complicate the equation, so we typically start by removing it in a controlled manner.

• Square both sides: To eliminate the square root, square both sides of the equation. This transforms a square root equation into a polynomial form, which is often easier to solve.
• Be cautious of extraneous solutions: Squaring both sides can introduce solutions that do not satisfy the original equation. It is crucial to verify potential solutions by substituting them back into the original equation.
• Maintain the equation's integrity: Ensure the operations are valid and do not alter the fundamental nature of the equation.

Understanding square root equations is essential for tackling algebraic problems involving surds.

Equation Verification

Equation verification is a crucial final step to ensure the solutions obtained are accurate and appropriate. Once you find a potential solution, substituting it back into the original equation can confirm its validity.

• Substitute and simplify: Replace the variable with the solution found and simplify both sides of the original equation.
• Check for equality: Ensure that both sides of the equation yield the same value after substitution. If they do, the solution is correct.
• Identify extraneous roots: This step is vital, especially after operations like squaring both sides of an equation, which can introduce false solutions.

Thorough verification helps avoid mistakes and ensures that the solution makes logical sense in the context of the problem.