Problem 18 - Intermediate Algebra

Question

Use a check to determine whether 5 is a solution of each equation. See Example 1. $$ \frac{3}{5} p-5=-2 $$

Step-by-Step Solution

Answer

Yes, 5 is a solution because it satisfies the equation.

Step 1: Substitute the Value

To determine if 5 is a solution to the equation \( \frac{3}{5} p - 5 = -2 \), substitute \( p = 5 \) into the equation. This gives us: \( \frac{3}{5} \times 5 - 5 = -2 \).

Step 2: Simplify the Expression

Simplify the left side of the equation. Calculate \( \frac{3}{5} \times 5 \). This results in 3, so the equation becomes \( 3 - 5 = -2 \).

Step 3: Verify the Equation

Simplify the expression \( 3 - 5 \) on the left side to see if it equals the right side. The result is \( -2 \), which matches the right side of the original equation \( -2 \).

Step 4: Draw a Conclusion

Since substituting \( p = 5 \) into the equation \( \frac{3}{5} p - 5 = -2 \) results in a true statement \( -2 = -2 \), 5 is indeed a solution of the equation.

Key Concepts

Solution Verification

When faced with an algebraic equation, verifying if a given number is a solution is an essential skill. This process ensures that the solution you propose makes the equation hold true. To verify the solution:

• Substitute the proposed number into the equation.
• Simplify the equation to check for equality on both sides.
• Conclude if the left-hand side equals the right-hand side.

To sum it up, if the two sides of the equation are equal after substitution, then the number is indeed a solution.

Substitution Method

The substitution method is a key technique in solving and verifying equations. It involves replacing a variable with a numeric value to transform the equation into a simpler form. Here's how it works:

• Identify the variable you need to substitute.
• Insert the numerical value given for the variable.
• Perform any calculations needed to further simplify the expression.

By substituting, you essentially fix one side of the equation to test if the entire expression holds true. This is a straightforward way to check if your solution is valid.

Simplifying Algebraic Equations

Simplifying equations is a fundamental part of solving equation problems. It helps to reduce complicated expressions into something more manageable, making it easier to verify solutions. Here's a quick guide:

• Perform arithmetic operations like addition, subtraction, multiplication, and division as needed.
• Combine like terms to simplify the equation further.
• Ensure to follow the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (often remembered as PEMDAS).

The goal of simplifying is to make it evident whether your substitution produces a true statement, helping you verify that a number is a solution of the equation.