Problem 11 - Prealgebra

Question

Solve each equation. Check your solution. $$9=\frac{3}{4} g$$

Step-by-Step Solution

Answer

The solution is \(g = 12\).

Step 1: Understand the Equation

The equation we need to solve is \(9 = \frac{3}{4}g\). Our goal is to find the value of \(g\) that satisfies this equation.

Step 2: Isolate the Variable

To find the value of \(g\), we need to get \(g\) by itself on one side of the equation. Currently, \(g\) is being multiplied by \(\frac{3}{4}\). We can isolate \(g\) by multiplying both sides of the equation by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). This gives us \(g = 9 \times \frac{4}{3}\).

Step 3: Simplify the Expression

Now let's simplify the expression we found: \(g = 9 \times \frac{4}{3}\). First, multiply \(9\) by \(4\), which is \(36\), then divide by \(3\) to get \(g = 12\).

Step 4: Check the Solution

To ensure our solution is correct, substitute \(g = 12\) back into the original equation: \(9 = \frac{3}{4} \times 12\). Calculate \(\frac{3}{4} \times 12 = 9\). Since both sides of the equation are equal, our solution \(g = 12\) is verified.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols to solve problems. In many cases, these symbols represent numbers, and algebra allows us to create equations to represent real-world scenarios mathematically. By using algebra, one can solve for unknown values that satisfy given conditions. For example, in the equation \( 9 = \frac{3}{4}g \), \( g \) is the unknown we need to find.The key to understanding algebra is learning how to rearrange equations. This involves working with the components of an equation - numbers, variables, and operators - to isolate the variable you're trying to solve. Algebra provides the tools needed to express and solve these types of equations.

Solving Equations

Solving equations is an essential skill in algebra. It involves finding the value of the variable that makes the equation true. To solve an equation, one must manipulate the equation to isolate the variable. For the equation \( 9 = \frac{3}{4}g \), our variable \( g \) is currently multiplied by the fraction \( \frac{3}{4} \).To isolate \( g \), we multiply both sides of the equation by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). This step eliminates the fraction, resulting in \( g = 9 \times \frac{4}{3} \).

• Multiply by reciprocal: This counteracts the division, effectively bringing all terms involving \( g \) on one side of the equation.
• Simplify calculations: Multiply 9 by 4, then divide the result by 3.

Following these steps gives us \( g = 12 \), the solution to our equation.

Verifying Your Solution

Once you have found a solution to an equation, it is essential to verify it to ensure it is correct. This means substituting the solution back into the original equation and checking if both sides of the equation are equal. This step is crucial because it confirms the accuracy of your work.For our equation \( 9 = \frac{3}{4}g \), we found \( g = 12 \). Substituting 12 for \( g \) in the equation gives us \( 9 = \frac{3}{4} \times 12 \). By calculating \( \frac{3}{4} \times 12 \), we get 9, making both sides of the equation equal.

• Substitution: Replace the variable with the value you found.
• Equality Check: Calculate both sides to see if they are the same.

This verification process confirms that \( g = 12 \) is indeed the correct solution, reinforcing the accuracy of your algebraic manipulation.