Problem 38 - Elementary Algebra

Question

Solve. $$ -3 x=54 $$

Step-by-Step Solution

Answer

x = -18.

Step 1: Understand the Equation

Given the linear equation \[ -3x = 54 \]The objective is to find the value of \( x \) that makes this equation true.

Step 2: Perform Division

To isolate \( x \), divide both sides of the equation by \(-3\).\[ x = \frac{54}{-3} \]

Step 3: Simplify the Expression

Simplify the right side of the equation.Calculate \( \frac{54}{-3} \), which equals \(-18\).So, \( x = -18 \).

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions representing a straight line when plotted on a graph. These equations typically look like \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable to be found. Solving these equations involves finding the value of \( x \) that makes the equation true. Understanding the purpose of solving linear equations helps you tackle real-world situations like finding distances, financial projections, or even scientific data computations!

Here's a simple approach to solving them:

• Identify the equation and recognize it's in the form of \( ax + b = c \).
• Your goal is to get \( x \) alone on one side of the equation to find its value.
• Follow necessary operations (addition, subtraction, multiplication, division) as required by the equation.

With practice, you will master these types of problems, quickly and intuitively.

Isolation of Variable

The key to solving for an unknown in an equation is isolating the variable. This means moving all terms involving \( x \) to one side so \( x \) stands alone. For our example problem, \( -3x = 54 \), the variable \( x \) is already isolated with its coefficient on one side, which in this case is \(-3\).

Steps to isolate the variable:

• Move constants and any other variables to the opposite side of \( x \) using operations like addition or subtraction.
• Perform the inverse of any operations affecting \( x \) directly. Here, divide by \(-3\), as it is multiplying with \( x \).

Isolation requires applying basic algebraic operations carefully to transform the equation to a simple form: \( x = \) some number thus finding your solution easily.

Division in Algebra

Division is fundamental in algebra for solving equations where terms must be simplified or canceled out. Our challenge, \( -3x = 54 \), uses division to simplify an equation that features multiplication.

Here's how division works in algebra:

• Identify terms in multiplication with the variable needing simplification.
• Divide each term on both sides of the equation by the number multiplying the variable. This ensures each side of the equation remains balanced.
• Simplify the division to find the value of the variable.

In the example, dividing both sides by \(-3\) gives us \( x = \frac{54}{-3} \). Simplifying this yields \(-18\). Remember, division in algebra ensures clarity and simplicity in solving equations.