Problem 33 - Precalculus: Functions and Graphs

Question

Solve the equation. \(\sqrt{7-5 x}=8\)

Step-by-Step Solution

Answer

The solution is \( x = -\frac{57}{5} \).

Step 1: Understand the equation

We are given the equation \( \sqrt{7 - 5x} = 8 \). This is a square root equation, and our task is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate the square root

Since the square root is already isolated on one side of the equation (\( \sqrt{7 - 5x} \)), we don't need any additional operations here.

Step 3: Eliminate the square root

Square both sides of the equation to eliminate the square root:\[ (\sqrt{7 - 5x})^2 = 8^2 \]This simplifies to:\[ 7 - 5x = 64 \]

Step 4: Solve for \( x \)

Now, solve the linear equation \( 7 - 5x = 64 \).First, subtract 7 from both sides:\[ 7 - 5x - 7 = 64 - 7 \]This simplifies to:\[ -5x = 57 \]Then, divide both sides by -5:\[ x = \frac{57}{-5} \]Therefore, \( x = -\frac{57}{5} \).

Step 5: Verify the solution

Plug \( x = -\frac{57}{5} \) back into the original equation to verify:\[ \sqrt{7 - 5\left(-\frac{57}{5}\right)} = \sqrt{7 + 57} = \sqrt{64} = 8 \]Since both sides equal 8, our solution is verified.

Key Concepts

Solving Linear Equations

Linear equations are expressions that involve variables with no exponents or powers higher than one. These types of equations are foundational in algebra and are typically solved in a few straightforward steps. In the context of the original problem, once the square root has been eliminated, the equation becomes linear. Here's the step-by-step approach to solving it:

• Firstly, identify all terms involving the variable on one side of the equation.
• Isolate the variable by performing inverse operations, which involves adding or subtracting terms to both sides to remove any constants from the variable side.
• Next, divide or multiply both sides of the equation by the coefficient of the variable to solve for it.

In the example, after squaring both sides, the equation becomes \[ 7 - 5x = 64 \]. Subtract 7 from both sides to isolate terms involving \( x \): \[ 7 - 7 - 5x = 64 - 7 \]. This simplifies to \[ -5x = 57 \]. Finally, divide both sides by -5 to solve for \( x \): \[ x = \frac{57}{-5} \], resulting in \( x = -\frac{57}{5} \). This straightforward approach to solving linear equations allows us to find the value of the unknown quickly and efficiently.

Eliminating Square Roots

Eliminating square roots helps to simplify equations, allowing us to work with linear forms instead. The most effective method to remove a square root is by squaring both sides of the equation. This must be done carefully to ensure the equivalence of the equation remains balanced and no extraneous solutions are introduced.

• Identify the square root expression. In the problem at hand, it is \( \sqrt{7 - 5x} \).
• Apply squaring as an inverse operation to eliminate the square root. Remember, squaring both sides should be applied correctly, ensuring the whole equation is encapsulated, maintaining equality.
• After squaring, simplify the result to extract a standard algebraic equation.

In the original exercise, squaring both sides led directly to \[ (\sqrt{7 - 5x})^2 = 8^2 \]. This simplifies to \[ 7 - 5x = 64 \], thus having successfully removed the square root and rendering the rest of the equation simpler to solve. This technique transforms the equation into a more manageable form, setting up easy transition to solving for \( x \).

Verification of Solutions

Verification of solutions is a crucial step in solving equations, ensuring that the value obtained for a variable indeed satisfies the original equation. It serves as a check for accuracy and helps avoid any errors due to computational mistakes or extraneous solutions introduced during the process of elimination.

• Substitute the solution back into the original equation. This involves replacing the variable with the derived solution.
• Simplify the equation and verify that both sides hold true, matching as initially presented in the problem statement.
• If both sides of the equation are equal upon substitution, the solution is verified.

In our example, after finding \( x = -\frac{57}{5} \), we substitute it back into the original equation: \[ \sqrt{7 - 5\left(-\frac{57}{5}\right)} = \sqrt{7 + 57} = \sqrt{64} \]. The left side simplifies to 8, which equals the right side, confirming that our solution is correct. Verification underscores the reliability of mathematical solutions, ensuring that the processes and results align with the initial problem statement.