Problem 17 - Intermediate Algebra

Question

Use a check to determine whether 5 is a solution of each equation. See Example 1. $$ 3(2 m-3)=15 $$

Step-by-Step Solution

Answer

\(m=5\) is not a solution since \(3(2m-3)\neq15\) when \(m=5\).

Step 1: Understand the Equation

The equation provided is \(3(2m-3)=15\). We need to determine if substituting \(m=5\) into this equation satisfies it.

Step 2: Substitute the Value

Substitute \(m=5\) into the equation: \[3(2(5)-3)=15\]Evaluate inside the parentheses first.

Step 3: Simplify Inside the Parentheses

Calculate \(2(5)\) which equals \(10\). Substitute back into the equation to get \[3(10-3)=15\]Now simplify \(10-3\).

Step 4: Calculate the Result

Compute \(10-3 = 7\), so the equation becomes \[3(7)=15\]. Simplify further by multiplying.

Step 5: Multiply

Now multiply \(3\) by \(7\):\[3 \times 7 = 21\] This does not equal \(15\).

Step 6: Conclude

Since \(21 eq 15\), substituting \(m=5\) into the equation does not satisfy it. Hence, \(m=5\) is not a solution.

Key Concepts

Solution Verification

When solving algebraic equations, one crucial task is ensuring a proposed solution is correct. In this exercise, we check whether substituting a specific value into the equation holds true. The process of solution verification involves substituting the value into the original equation and determining if the equality remains valid.
To verify, follow these steps:

• Substitute the given value into the equation.
• Simplify and solve both sides.
• Check if both sides of the equation are equal.

If both sides are not equal, the proposed value is not a solution. Verification is essential because it confirms the accuracy of your answer.

Substitution Method

The substitution method in algebra involves replacing a variable with a given value to evaluate an expression or equation.
In this exercise, the substitution method is used to check if 5 is a solution to the equation \(3(2m - 3) = 15\). By substituting \(m=5\) into the equation, we evaluate how the equation behaves with this specific value.
To do this effectively:

• Replace the variable with the given number.
• Perform arithmetic operations as they appear.

This method provides a step-by-step approach to understand how each part of the equation responds to the substituted value, making it a powerful tool for verifying solutions.

Simplifying Expressions

Simplifying expressions is key in solving algebraic equations, especially when verifying if a value like 5 is a solution. Simplifying involves performing arithmetic operations, such as addition, subtraction, multiplication, and division while following the order of operations (PEMDAS/BODMAS).
In the exercise, the simplification process involved calculating two main steps:

• First compute inside the parentheses: \(2(5) - 3\).
• Then multiply by 3 to verify the equation: \(3(7)\).

After simplification, if both sides of the equation no longer equal, it indicates that the value substituted is not a solution. Simplification ensures complex expressions are reduced to simpler forms, allowing for straightforward comparison and verification.