Problem 16 - Intermediate Algebra

Question

Use a check to determine whether 5 is a solution of each equation. See Example 1. $$ 7 x-2=53-5 x $$

Step-by-Step Solution

Answer

5 is not a solution of the given equation.

Step 1: Substitute the Given Value

To check if 5 is a solution, substitute \( x = 5 \) into the equation: \[ 7(5) - 2 = 53 - 5(5). \]

Step 2: Calculate Left Side of Equation

First, calculate the left side of the equation: \[ 7(5) - 2 = 35 - 2 = 33. \]

Step 3: Calculate Right Side of Equation

Next, calculate the right side of the equation: \[ 53 - 5(5) = 53 - 25 = 28. \]

Step 4: Compare Both Sides

Now, compare the left side and the right side: The left side is 33 and the right side is 28. Since 33 does not equal 28, \( x = 5 \) is not a solution.

Key Concepts

Understanding Solving Algebraic Equations

When it comes to solving algebraic equations, the main goal is to determine the value of the variable that makes the equation true. An equation is a statement of equality between two expressions. For instance, in the equation \( 7x - 2 = 53 - 5x \), you need to find the value of \( x \) that satisfies both sides of the equation.
By solving, you mean isolating the variable on one side to find its value. This helps in understanding how quantities relate to each other, which is fundamental in algebra.

• You'll often perform operations like addition, subtraction, multiplication, or division on both sides of the equation to maintain balance.
• The solution is the number that you substitute back into the original equation to verify its correctness.

Identifying when both sides are equal is key to confirming you've found the correct solution.

Exploring the Substitution Method

The substitution method is a straightforward approach to determining if a particular value is a solution to an equation. It involves replacing the variable in the equation with a given number and simplifying both sides to see if they equal. For the example equation \( 7x - 2 = 53 - 5x \), substitution is used to check if \( x = 5 \) is a solution.
The process is simple:

• Begin by substituting the suggested value of the variable into both sides of the equation. Here, replace \( x \) with 5.
• Simplify the expressions on both sides to calculate their values.
• Compare the two results. If they are equal, then the substituted value is indeed a solution.

It's an effective method for checking specific solutions, but remember, until both sides match, the solution isn't valid.

Evaluating Solution Verification

Solution verification is the last step where you check whether your proposed solution actually works in the original equation. It's a way to ensure that errors weren't made during calculations.
In the example problem, after substituting \( x = 5 \) into the equation and simplifying:

• The left side calculated to \( 33 \).
• The right side calculated to \( 28 \).
• Since \( 33 eq 28 \), \( x = 5 \) is not a valid solution.

This step is crucial, as it confirms that your solution satisfies the original equation. Without this step, there might be uncertainties lurking in your result. Always substitute back the solution into the equation to verify accuracy.