Problem 22 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ \frac{1}{5} x-1=-\frac{4}{5} x-13 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -12 \).

Step 1: Simplify Both Sides

The given equation is \( \frac{1}{5} x - 1 = -\frac{4}{5} x - 13 \). To simplify, consider the terms with \( x \) and constants on both sides. Currently, each side is already simplified with no like terms to combine further.

Step 2: Eliminate Fractions by Multiplying

To remove the fractions, multiply every term in the equation by 5 (the denominator). This gives: \( 5 \times \left( \frac{1}{5} x \right) - 5 \times 1 = 5 \times \left( -\frac{4}{5} x \right) - 5 \times 13 \), simplifying to: \( x - 5 = -4x - 65 \).

Step 3: Isolate the Variable

Add \( 4x \) to both sides to get all terms with \( x \) on one side: \( x + 4x - 5 = -4x + 4x - 65 \). Simplifying further, we have: \( 5x - 5 = -65 \).

Step 4: Solve for the Variable

Add 5 to both sides to isolate the term with \( x \): \( 5x - 5 + 5 = -65 + 5 \), resulting in \( 5x = -60 \). Divide both sides by 5 to solve for \( x \): \( 5x / 5 = -60 / 5 \), so \( x = -12 \).

Step 5: Verify the Solution

Substitute \( x = -12 \) back into the original equation to check: \( \frac{1}{5}(-12) - 1 \) should equal \( -\frac{4}{5}(-12) - 13 \). Calculating, \( \frac{1}{5}(-12) = -\frac{12}{5} \) and \( -1 = -5/5 \), so the left side is \( -\frac{17}{5} \). For the right side, \( -\frac{4}{5}(-12) = \frac{48}{5} \), and subtracting 13 gives \( \frac{48}{5} - \frac{65}{5} = -\frac{17}{5} \). The sides match, confirming \( x = -12 \) is correct.

Key Concepts

Fractions in Equations

Solving equations with fractions can initially seem intimidating, but it's all about understanding their behavior and removing them for simplicity. When you see fractions in equations, as in the example \( \frac{1}{5}x - 1 = -\frac{4}{5}x - 13 \), you might want to remove them to ease the process. This step is crucial as it leads to a clearer, fraction-free equation.

Here's a simple way to handle fractions:

• Identify the least common denominator of all fractions in the equation.
• Multiply every term in the equation by this common denominator. This transforms fractional coefficients into whole numbers.

In the given equation, multiplying by 5, which is the denominator, simplifies it to \( x - 5 = -4x - 65 \). This step alone makes the equation much easier to manage, removing complexities and reducing calculation errors. Remember, the key is ensuring each term is affected equally, maintaining the equation's balance while saying goodbye to fractions.

Variable Isolation

Variable isolation is like playing a game of keep-away with the other numbers in the equation. The goal is to get the variable (like \( x \) in our example) alone on one side of the equation. Let's break down how to do this effectively.

Once you've cleared out any fractions, your task is straightforward:

• Gather all terms containing the variable on one side of the equation. This often involves adding or subtracting terms from both sides.
• For equations like \( x - 5 = -4x - 65 \), start by adding \( 4x \) to both sides to avoid negative coefficients, simplifying to \( 5x - 5 = -65 \).
• Next, eliminate any constants on the variable's side by adding or subtracting the same value from both sides, easily resulting in \( 5x = -60 \).

By isolating the variable, you're basically simplifying the pathway to the solution. It's like peeling layers off an onion, removing distractions one step at a time until only the variable and its coefficient remain on one side.

Equation Simplification

Simplification is the final flourish in the equation solving process, allowing you to solve for the variable efficiently. Think of it as tidying up after you've moved everything where it needs to be. Once you isolate the variable, simplify further by performing arithmetic operations to solve for the variable.

In our example, once we reach the equation \( 5x = -60 \), we simplify it:

• Divide both sides by the coefficient of the variable (here, it is 5). This is crucial as it turns your equation into \( x = -12 \).
• By performing such simplification, you ensure the variable is expressed in the simplest form possible.

This step involves straightforward arithmetic, transforming the equation from its complex beginning into an understandable result for the variable. Moreover, once you've simplified, don't forget to double-check by plugging this solution back into the original equation to verify correctness. This reaffirms your final answer and ensures no errors slipped through during your calculations.