Problem 32 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$\frac{8 x}{9}=\frac{2}{3}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{3}{4}\).

Step 1: Isolate the fraction

To isolate \(x\), multiply both sides of the equation by \(\frac{9}{8}\), which is the reciprocal of the coefficient of \(x\). This gives: \[x = \frac{2}{3} \cdot \frac{9}{8}\]

Step 2: Simplify the equation

Solving the right side, we get: \[x = \frac{3}{4}\] This is the solution to the equation.

Step 3: Check the solution

Substitute \(x = \frac{3}{4}\) back into the original equation and simplify to check if both sides of the equation are equal: \[\frac{8 \cdot \frac{3}{4}}{9} = \frac{2}{3}\] simplifying, we get: \[\frac{2}{3} = \frac{2}{3}\] Both sides of the equation are equal, so \(x = \frac{3}{4}\) is the solution.

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole as numbers. They consist of a numerator and a denominator, where the numerator is the top number indicating the number of equal parts being considered, and the denominator is the bottom number showing the total number of equal parts in the whole. For example, in the fraction \( \frac{8}{9} \), 8 is the numerator and 9 is the denominator.
Fractions are used in solving equations to represent ratios or proportions. Understanding how to manipulate fractions, such as finding a common denominator or simplifying them, is crucial when working with algebraic equations. When solving equations with fractions, it's often necessary to isolate the variable on one side to find the solution. This usually involves performing operations such as multiplication or division with the reciprocal of the fraction containing the variable.

The Importance of Checking Solutions

Checking solutions is a crucial step in solving equations. It ensures that the value you found for the variable actually satisfies the original equation. Once you’ve found a potential solution, substitute it back into the original equation.
For instance, after finding \( x = \frac{3}{4} \) in our problem, substituting it back we should perform calculations like multiplying and simplifying fractions to verify that both sides of the equation remain equal. This step confirms the solution or helps identify mistakes that might have been made during the manipulation steps. If both sides are equal when the solution is substituted into the original equation, then the solution is correct. If they aren’t, something likely went wrong, and you’ll need to revisit your steps.

Mastering Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to simplify, rearrange, or solve equations. It's a key skill when working to isolate a variable. One essential technique is using the reciprocal of a fraction to eliminate the fraction when a variable appears in one.
In the given problem, to isolate \( x \), we multiply both sides by \( \frac{9}{8} \), which is the reciprocal of \( \frac{8}{9} \). This operation effectively cancels the fraction and makes it easier to solve for the variable. Additionally, simplifying expressions by canceling common factors on both the numerator and the denominator helps in clearly seeing the solution. Algebraic manipulation also includes re-arranging terms and factoring, which are useful techniques to master for tackling complex equations.