Problem 22 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ -6 v(2 t-9)=0 $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{9}{2} \).

Step 1: Identify the Multiplicative Property of Zero

Recall that any expression multiplied by 0 results in 0. Therefore, if \[-6v(2t-9) = 0,\]either \(-6v\) or \((2t-9)\) must equal 0.

Step 2: Set the Factor to Zero

To solve the equation, set the factor that contains the variable to zero. Here, since \[-6v = 0\]could be used, let us focus on \[(2t-9) = 0,\]as this contains another variable which is generally more useful for solutions.

Step 3: Solve the Equation for t

Set the factor \((2t-9)\) equal to zero to solve for \(t\):\[(2t - 9) = 0.\]Add 9 to both sides:\[2t = 9.\]Next, divide by 2 to isolate \(t\):\[t = \frac{9}{2}.\]

Key Concepts

Multiplicative Property of Zero

The multiplicative property of zero is a fundamental rule in mathematics. It states that any number or expression multiplied by zero equals zero. This is crucial for solving equations since it helps simplify and break down the problem. For instance, if we have

• \[-6v(2t-9) = 0,\]

it implies either \(-6v\) or \((2t-9)\) must equal zero to satisfy the equation. Identifying this property quickly allows you to focus on solving each contributing factor separately. This concept ensures all parts equal zero, reinforcing why factors contribute to the outcome when multiplied.

Factor Equation

When faced with an equation, factoring is a helpful technique for finding solutions. 'Factoring' means breaking down an equation into simpler, multiplying parts or factors. For the equation

• \[-6v(2t-9)=0,\]

we recognize the term as a product. Factoring involves assuming either \(-6v = 0\) or \((2t-9) = 0\) to solve for unknown variables. This approach leverages the multiplicative property of zero to isolate parts of the equation that need to be addressed separately. Factoring facilitates solving by narrowing down which factor might successfully equal zero, moving the solution process forward effectively.

Isolating Variables

Isolating variables is a strategic step in solving equations. This process involves manipulating the equation to have the variable on one side, making it easier to determine its value. In the equation

• \[(2t - 9) = 0,\]

we start by adding 9 to both sides:

• \[2t = 9.\]

This change helps in bringing the variable 't' by itself. Next, divide by 2:

• \[t = \frac{9}{2}.\]

This straightforward approach is crucial for solving algebraic equations, allowing us to find solutions efficiently by dealing with one variable at a time.