Problem 18 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{4}{5} x-\frac{8}{5}=-\frac{16}{5} $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Eliminate Fractions

First, we eliminate the fractions by multiplying every term in the equation by 5, the common denominator. Doing so, we get:\[ 5 \times \left( \frac{4}{5} x \right) - 5 \times \left( \frac{8}{5} \right) = 5 \times \left( -\frac{16}{5} \right) \]This simplifies to: \[ 4x - 8 = -16 \]

Step 2: Isolate the Variable Term

Next, we want to get the variable term by itself on one side of the equation. We will add 8 to both sides of the equation to cancel the -8 on the left side:\[ 4x - 8 + 8 = -16 + 8 \]This simplifies to:\[ 4x = -8 \]

Step 3: Solve for x

Now, to solve for \( x \), divide each side of the equation by 4:\[ \frac{4x}{4} = \frac{-8}{4} \]This gives us:\[ x = -2 \]

Key Concepts

Solving Linear Equations

Linear equations are mathematical statements that set two expressions equal to each other, involving variables and constants. Solving these equations means finding the value of the variable that makes the equation true.

A linear equation often appears as a simple algebraic expression (like ax + b = c), where we aim to find the value of "x" that balances both sides. The overall goal in solving a linear equation is to "isolate" the variable (often represented as "x") on one side of the equation. This helps us discover its value.

The process involves a series of straightforward algebraic steps intended to simplify the equation into the form "x = value". These steps often include fraction elimination, variable isolation, and simplifying arithmetic operations. Keeping each side of the equation balanced is crucial throughout the process.

Fraction Elimination

Before tackling the rest of the equation, it is often helpful to eliminate fractions. This simplification can make the problem much easier to handle.

To eliminate fractions, identify a common denominator across all terms, then multiply each term on both sides of the equation by this denominator. For example, in the exercise, multiplying every term by 5—the common denominator—removes fractions:

• \( \frac{4}{5}x - \frac{8}{5} = -\frac{16}{5} \) transforms to \( 4x - 8 = -16 \).

This approach leaves you with an equation involving whole numbers, which are far easier to work with.

Variable Isolation

Isolating the variable of interest is central to solving linear equations. Once fractions are eliminated, the next step is to rearrange the equation to have the variable term on one side alone.

For example, to isolate \( x \) in \( 4x - 8 = -16 \), add 8 to both sides:

• \( 4x - 8 + 8 = -16 + 8 \) simplifies to \( 4x = -8 \).

This step effectively transfers constants from one side to another, focusing on the variable's value without interference from other numbers.

Effective isolation sets you up to solve the final part of the equation easily.

Basic Algebra Steps

Basic algebra involves a series of ordered steps to solve equations efficiently.

These steps usually start with simple manipulations, like fraction elimination, followed by carefully isolating terms and finally solving for the variable. Each step needs to maintain the equality by performing the same operation on both sides.

• In the exercise's last step, dividing both sides by 4 gives \( x \) its own value: \( \frac{4x}{4} = \frac{-8}{4} \) yields \( x = -2 \).

It’s essential to keep equations balanced by "doing unto one side what you do to the other." This practice helps ensure the solution is correct and the equation remains valid throughout the process.

Mastery of these steps underpins all of algebra, empowering you to tackle more complex problems with confidence.