Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

In your own words, describe how to solve a linear equation.

Step-by-Step Solution

Answer

To solve a linear equation: 1. Identify and understand the equation. 2. Isolate the variable by adding or subtracting terms which do not involve the variable, to the other side of the equal sign. 3. After isolating, solve for the variable by dividing or multiplying both sides of the resulting equation with the coefficient of the variable.

Step 1: Understand the Equation

A linear equation typically takes the form \( ax + b = 0 \), where a and b are constants, and x is the variable you must solve for. The first step will be to understand the equation, recognizing the values of a, b and identify the missing variable - x.

Step 2: Isolate the Variable

The next step in solving a linear equation is to isolate the variable. One must change the equation format to \( x = something \). This is done by moving all the terms except the term with 'x' to the opposite side of the equation. This process involves either addition or subtraction operation.

Step 3: Solve for the Variable

Final step involves simplifying the equation by performing the appropriate mathematical operations to solve for x. If the term with 'x' involves multiplication or division, apply the reverse operation to both sides of the equation to get the value of x.

Key Concepts

Linear Equation

Imagine a seesaw balanced perfectly, each side holding equal weight. A linear equation is much like this seesaw, maintaining a balance of quantities on both sides of the equal sign. In its simplest form, it looks like \( ax + b = c \) where \( a \) and \( b \) are numbers, and \( x \) is the variable we want to find, much like the unknown weight on one side of the seesaw. When you hear 'linear,' think line—because if you plot the solutions to these equations, they'll always form a straight line on a graph.

But why call it an equation? Because it equates two things: the expression on the left side is valued the same as the expression on the right side. And to 'solve' this equation means to find the value of \( x \) that makes the expression on both sides of the equation equal. This process of finding \( x \) is like making sure both sides of the seesaw are balanced.

Isolate the Variable

The key step to solving a linear equation is to isolate the variable, which means to get the variable \( x \) by itself on one side of the equation. Picture a balance scale: you can add or remove the same weight from both sides without disturbing the balance. In the same way, whatever you do to one side of the equation, you must do to the other to keep it equal. If \( x \) is mingled with other numbers, we need to move those companions to the opposite side. This is usually done by doing the opposite operation: if \( x \) is being added by 3, we subtract 3 from both sides to cancel it out on the side with \( x \) and thus isolate \( x \) effectively.

Mathematical Operations

Adding and Subtracting

Equations are all about balance, and mathematical operations are the tools we use to maintain it. We can add or subtract the same number on both sides of the equation. For example, if you see \( x + 5 = 12 \) and you want to get rid of the 5, simply subtract 5 from both sides resulting in \( x = 7 \).

Multiplying and Dividing

If \( x \) is multiplied by a number, we can divide both sides by that number to isolate \( x \)—like untying a knot. Conversely, if \( x \) is divided by a number, we can multiply both sides by that same number to free \( x \) from the divide. So for the equation \( 3x = 9 \), we'd divide by 3 on both sides, leaving us with \( x = 3 \).

Algebraic Methods

Solving a linear equation is like following a recipe; it's a process that involves steps called algebraic methods. These methods give structure to our approach in finding variable values. Let's think about our starting equation \( ax + b = 0 \). First, we aim to eliminate \( b \) by subtracting it from both sides, because it's not attached to the \( x \). This simplifies to \( ax = -b \). Next, to get \( x \) alone, we divide each side by \( a \) (as long as \( a \) isn't zero), leading us to \( x = -\frac{b}{a} \). Voila! We've isolated \( x \) using algebraic manipulation, effectively applying our algebraic methods to balance the equation and find the value of our unknown.