Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ \sqrt{7-5 x}=8 $$

Step-by-Step Solution

Answer

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

Step 1: Remove the Square Root

To eliminate the square root, square both sides of the equation. This means we take the expression under the square root, \(7-5x\), and square the right side, \(8\), as well. The equation becomes: \[(\sqrt{7 - 5x})^2 = 8^2\] Simplifying, this gives:\[7 - 5x = 64\]

Step 2: Isolate the Variable Term

Next, to solve for \(x\), begin by isolating the term containing \(x\). Subtract 7 from both sides of the equation:\[7 - 5x - 7 = 64 - 7\]Simplifying gives:\[-5x = 57\]

Step 3: Solve for x

Finally, to solve for \(x\), divide both sides of the equation by -5:\[-5x / -5 = 57 / -5\]Simplifying results in:\[x = -\frac{57}{5}\]

Key Concepts

Square Roots

When looking at the equation \( \sqrt{7 - 5x} = 8 \), the square root symbol is crucial to understand. A square root asks, "What number multiplied by itself gives this number?" In this equation, we have \( \sqrt{7 - 5x} \), which means we are looking for a number that, when squared, equals \( 7 - 5x \).

To simplify or eliminate a square root, you can square both sides of the equation. Squaring "undoes" the square root, effectively removing it. This process transforms the equation. For example:

• Original equation: \( \sqrt{7 - 5x} = 8 \)
• Squaring both sides: \((\sqrt{7 - 5x})^2 = 8^2\)
• This gives: \(7 - 5x = 64\)

Understanding square roots and squaring are essential skills in algebra because they help in simplifying and solving equations efficiently.

Equation Solving Process

Solving an equation is like solving a puzzle, where each step brings you closer to the solution. It's essential to approach it systematically.

Here's a brief guide to the equation solving process:

• Identify the operation: Recognize operations used in the equation (e.g., addition, multiplication, square roots). This helps in determining what steps will "undo" those operations.
• Apply the inverse operation: To cancel square roots, squares, etc., use inverse operations. For square roots, squaring is the inverse operation.
• Simplify: After applying inverse operations, simplify the equation to make it more manageable.

In our example, we first handled the square root by squaring both sides. Then, the equation was simplified to \( 7 - 5x = 64 \), which set the stage for isolating and solving for \( x \). This methodical approach ensures clarity and accuracy.

Variable Isolation

Variable isolation means getting the variable by itself on one side of the equation. This helps to find its value easily.

In our example, the term with \( x \) is \( -5x \). Here's how we isolated \( x \):

• Subtract: We started with \( 7 - 5x = 64 \). By subtracting 7 from both sides, we eliminated the 7 on the left. This results in \( -5x = 57 \).
• Divide: Next, we needed to undo the multiplication by -5. So, we divided both sides by -5: \( x = -\frac{57}{5} \).

By following these steps, you achieve the isolation of \( x \), making it easier to see its value. This is a common technique in algebra, used to solve for unknowns accurately.