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Chapter 5: Problem 13

Solve each equation, and check the solutions. $$ (2 m-7)(m-3)=0 $$

Short Answer

Expert verified
The solutions are \( m = 3 \) and \( m = \frac{7}{2} \).

Step by step solution

01

Apply the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. For the equation \( (2 m - 7)(m - 3) = 0 \), set each factor equal to zero: \[ 2m - 7 = 0 \] \[ m - 3 = 0 \]
02

Solve the First Equation

Solve the first equation \( 2m - 7 = 0 \) by isolating \( m \):\r \[ 2m - 7 = 0 \] \[ 2m = 7 \] \[ m = \frac{7}{2} \]
03

Solve the Second Equation

Solve the second equation \( m - 3 = 0 \): \[ m - 3 = 0 \] \[ m = 3 \]
04

Check the Solutions

Substitute \( m = 3 \) and \( m = \frac{7}{2} \) back into the original equation to verify:\rFor \( m = 3 \):\r \[ (2(3) - 7)(3 - 3) = (6 - 7)(0) = (-1)(0) = 0 \] \rFor \( m = \frac{7}{2} \): \r \[ (2(\frac{7}{2}) - 7)(\frac{7}{2} - 3) = (7 - 7)(\frac{7}{2} - 3) = (0)(\frac{1}{2}) = 0 \] \rBoth values satisfy the original equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Product Property
The Zero Product Property is a fundamental concept in algebra. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. This principle is used to solve equations of the form \(ab = 0\). For instance, in the equation \( (2m-7)(m-3)=0\), we apply the Zero Product Property by setting each factor equal to zero: \( 2m - 7 = 0 \) and \( m - 3 = 0 \). This approach simplifies the problem and allows us to solve for the variable step by step.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. Let's look at our example again:\r\r
  • For the equation \(2m - 7 = 0\):
First, isolate the variable \(m\) by adding 7 to both sides of the equation: \[ 2m - 7 + 7 = 0 + 7 \] Simplifying, we get: \[ 2m = 7 \] Next, divide both sides by 2: \[ m = \frac{7}{2} \]\r\r
  • For the equation \(m - 3 = 0\):
Simply add 3 to both sides: \[ m = 3 \] These steps show how to isolate the variable to find its value.
Verifying Solutions
Verifying solutions ensures that the values we find actually satisfy the original equation. To check our solutions \( m = 3 \) and \( m = \frac{7}{2} \), substitute these values back into the original equation \( (2m-7)(m-3) = 0 \): \r\r
  • For \( m = 3 \):
\[ (2(3) - 7)(3 - 3) = (6 - 7)(0) = (-1)(0) = 0 \] This shows that \( m = 3 \) is a solution.\r\r
  • For \( m = \frac{7}{2} \):
\[ (2(\frac{7}{2})- 7)(\frac{7}{2} - 3) = (7 - 7)(\frac{1}{2}) = (0)(\frac{1}{2}) = 0 \] This shows that \( m = \frac{7}{2} \) is also a solution. Both values satisfy the original equation, confirming their correctness.

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