Problem 22 - Elementary Algebra

Question

For the following problems, solve the square root equations. $$ \sqrt{5 x+6}=\sqrt{3 x+7} $$

Step-by-Step Solution

Answer

Question: Solve the equation for x: $\sqrt{5x + 6} = \sqrt{3x + 7}$. Answer: $x = \frac{1}{2}$.

Step 1: Square both sides of the equation

To remove the square roots from the equation, we need to square both sides. On the left side, we square the square root of \((5x + 6)\), and on the right side, we square the square root of \((3x + 7)\). Doing so gives us: $$ (5x + 6) = (3x + 7) $$

Step 2: Solve for x

Now that we have a simplified equation without square roots, we can easily solve for x. Begin by isolating the variable x: Subtract \(3x\) from both sides: $$ 2x + 6 = 7 $$ Now, subtract \(6\) from both sides: $$ 2x = 1 $$ Finally, divide both sides by \(2\) to isolate the variable x: $$ x = \frac{1}{2} $$

Step 3: Check the solution

In order to be sure that \(\frac{1}{2}\) is a valid solution, we need to check that it satisfies the original equation. Plug the value of x back into the equation and verify that both sides are equal. $$ \sqrt{5 \left(\frac{1}{2}\right) + 6} = \sqrt{3 \left(\frac{1}{2}\right) + 7} $$ Simplify: $$ \sqrt{\frac{5}{2} + 6} = \sqrt{\frac{3}{2} + 7} $$ Find a common denominator to add the fractions: $$ \sqrt{\frac{17}{2}} = \sqrt{\frac{17}{2}} $$ Since both sides are equal, the solution \(x = \frac{1}{2}\) is correct.

Key Concepts

Solving Equations

Solving square root equations involves a few straightforward steps. The key is to begin by eliminating the square roots, which often form a barrier to direct solution. In our example, we have the equation \( \sqrt{5x + 6} = \sqrt{3x + 7} \). To simplify this, we square both sides of the equation. This effectively cancels the square roots, leaving us with the equation \( 5x + 6 = 3x + 7 \). This process might seem simple, but it ensures that we are working with values that can be isolated and manipulated easily. Remember, when you square an equation, solve both sides to maintain equality and ensure valid solutions.

Algebraic Manipulation

Algebraic manipulation is all about rearranging and simplifying equations to isolate the variable of interest. Once we square both sides and arrive at \( 5x + 6 = 3x + 7 \), we need to solve for \( x \). Begin by performing operations to move all terms involving \( x \) to one side and constant terms to the other. Subtract \( 3x \) from both sides to get \( 2x + 6 = 7 \). Afterward, subtract \( 6 \) from each side to simplify the equation further to \( 2x = 1 \). Finally, divide both sides by \( 2 \) to solve for \( x \), giving \( x = \frac{1}{2} \). This step-by-step manipulation is essential to solve equations efficiently and accurately.

Checking Solutions

Checking solutions is a vital step often overlooked. It's essential because squaring both sides during our solution process might introduce extraneous solutions. To verify our solution, \( x = \frac{1}{2} \), we substitute it back into the original equation: \( \sqrt{5 \left(\frac{1}{2}\right) + 6} = \sqrt{3 \left(\frac{1}{2}\right) + 7} \). Simplifying both sides results in \( \sqrt{\frac{17}{2}} = \sqrt{\frac{17}{2}} \). Because both sides are equal, \( x = \frac{1}{2} \) is indeed the correct solution. Always check your solutions to confirm their validity, especially when dealing with square roots, to ensure no extraneous solutions have been introduced.