Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$-x=17$$

Step-by-Step Solution

Answer

The solution to the equation \(-x = 17\) is \(x = -17\).

Step 1: Isolate the variable

Given the equation \(-x = 17\), start by multiplying the both sides by -1 to isolate \(x\), using the multiplication property of equality. This gives: \((-1) \cdot -x = (-1) \cdot 17\)

Step 2: Simplify the equation

After performing the multiplication, the equation becomes: \(x = -17\)

Step 3: Check the proposed solution

Substitute the proposed solution \(x=-17\) into the original equation \(-x=17\) to ensure it is valid. This gives: \(-(-17) = 17\), simplifying to \(17 = 17\), hence the solution is valid.

Key Concepts

Solving Linear Equations

Linear equations form the basis of algebra and appear extensively in various mathematical applications. Solving a linear equation involves finding the value of the variable that makes the equation true. To solve an equation like \( -x = 17 \) we need to perform operations that will get the variable \( x \) by itself on one side of the equation. This process is fundamental in algebra and prepares students for more complex problems down the line.

When solving linear equations, each operation performed must maintain the equation's balance by applying it to both sides. Understanding the properties of equality is crucial here, as it allows students to manipulate equations with confidence, knowing their operations are mathematically sound. The goal is always to simplify the equation to the form of \( x = \) some number, which is the solution to the equation.

Isolating the Variable

Isolating the variable is a key step in solving linear equations. It means rearranging the equation so that the variable we're solving for stands alone on one side of the equation. For the given equation \( -x = 17 \) we isolate \( x \) by multiplying each side by -1. This application of the multiplication property of equality is a strategic move to eliminate the negative sign attached to \( x \).

The simplification works like this: \[(-1) \cdot -x = (-1) \cdot 17\] resulting in \(x = -17\). It's a clean and effective method to clearly identify the value of the variable. Students should become comfortable with recognizing when and how to use this property to simplify equations efficiently.

Checking Solutions

Once a proposed solution has been found, it is imperative to verify its accuracy by checking the solution. This means substituting the value back into the original equation to ensure it holds true. For the equation \( -x = 17 \) with the solution \( x = -17 \) we check by plugging \( -17 \) back in for \( x \) and computing: \[ -(-17) = 17 \] which simplifies to \( 17 = 17 \).

By confirming the left side equals the right after substitution, we validate our solution. Checking is a critical step to avoid mistakes and to ensure understanding. It reinforces the concept that a true solution to an equation will always satisfy the original equation. It's a technique that also boosts confidence in students as they can self-assess their work for errors.