Problem 33 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$\frac{2}{3} x-4=6$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 15 \).

Step 1: Isolate the Variable Term

The given equation is \( \frac{2}{3}x - 4 = 6 \). To simplify it, we'll start by isolating the term containing the variable \( x \). Add 4 to both sides of the equation to remove the constant term on the left side. This results in: \[ \frac{2}{3}x - 4 + 4 = 6 + 4 \] Simplifying both sides, we have: \[ \frac{2}{3}x = 10 \]

Step 2: Solve for x

Now, we need to get \( x \) by itself. To do this, multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This will cancel out the fraction. \[ \frac{3}{2} \times \frac{2}{3}x = 10 \times \frac{3}{2} \] Simplifying both sides gives: \[ x = 15 \]

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental aspect of algebra. A linear equation is an equation of the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we need to find. The primary goal when solving linear equations is to find the value of \( x \) that makes the equation true. The process typically involves performing operations to both sides of the equation in such a way that we isolate the variable \( x \).

Steps to solve linear equations generally include:

• Eliminating any constant terms from the side of the equation where the variable resides.
• Dealing with any coefficients that are attached to the variable, often fractions or whole numbers.

For our given equation \( \frac{2}{3}x - 4 = 6 \), the first step involves using the addition property of equality to eliminate \(-4\), helping us focus on the \( \frac{2}{3}x \) term. After simplifying, isolate the variable further as needed.

Rational Coefficients

Rational coefficients are simply coefficients that are fractions. In a linear equation like \( \frac{2}{3}x - 4 = 6 \), the \( \frac{2}{3} \) is our rational coefficient. Working with rational coefficients involves some specific steps to simplify the equation effectively.

Here are some key points to remember:

• Rational coefficients can make the equation appear more complicated, but with proper operations, they are easy to manage.
• To eliminate a fraction, it's often useful to multiply by its reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This step simplifies the equation significantly.
• Understanding and working with fractions is an essential part of dealing with rational coefficients.

Our example used multiplication by the reciprocal, \( \frac{3}{2} \), to clear the fraction from \( \frac{2}{3}x \), simplifying the process of solving for \( x \).

Isolating Variables

Isolating the variable is a crucial step in solving any linear equation. The idea is to rearrange the equation so that \( x \) stands alone on one side, allowing us to see its value directly. This step is usually the final one, bringing the variable to one side of the equation and all other variables and constants to the other side.

To effectively isolate \( x \), follow these strategies:

• Remove any terms on the same side as \( x \) that do not contain \( x \). This is done by using the addition or subtraction property of equality.
• Next, handle any coefficients by dividing or multiplying to ensure \( x \) is by itself.
• Make sure to perform the same operation on both sides of the equation to maintain equality.

In the equation \( \frac{2}{3}x = 10 \), isolating \( x \) involved multiplying by \( \frac{3}{2} \), which eliminated the fraction and left \( x \) alone. This simplification helps us to find \( x = 15 \), our final solution.