Solving Linear Equations: Detailed Steps and Multiple Examples

This article provides comprehensive guidance on solving linear equations, utilizing multiple examples to illustrate the process. Linear equations are a fundamental part of algebra, and mastering them is essential for solving a wide range of mathematical problems.

Introduction to Linear Equations

A linear equation is any equation that can be written in the form of Ax B C, where A, B, and C are constants and x is the variable. Linear equations always have a solution, which can be found by isolating the variable on one side of the equation. This process involves a series of algebraic manipulations that maintain the equality of the equation.

Isolating the Variable - The Key to Solving Linear Equations

Let's start with a simple linear equation and take it through the steps to reach the solution.

Example 1: 4x - 3 17

To solve the equation 4x - 3 17, we need to isolate x. Here are the steps:

First, add 3 to both sides of the equation to move all constants to the right side: 4x - 3 3 17 3 Simplify to: 4x 20 Next, divide both sides by 4 to isolate x: 4x ÷ 4 20 ÷ 4 Simplify to: x 5 Finally, we can verify our solution by plugging x 5 back into the original equation: 4(5) - 3 17 20 - 3 17 Satisfies the equation, so x 5 is correct.

This process is a clear example of the fundamental algebraic principle of maintaining equality by performing the same operations on both sides of the equation.

Example 2: 4x - 5 17 - x

Let's solve 4x - 5 17 - x.

First, add x to both sides to move all variables to one side and all constants to the other: 4x - 5 x 17 - x x Simplify to: 5x - 5 17 Next, add 5 to both sides to isolate the term with x: 5x - 5 5 17 5 Simplify to: 5x 22 Finally, divide both sides by 5 to solve for x: 5x ÷ 5 22 ÷ 5 Simplify to: x 22/5 4.4 Verify by substituting x 4.4 into the original equation: 4(4.4) - 5 17 - 4.4 17.6 - 5 17 - 4.4 12.6 12.6

This confirmation ensures that the solution is correct.

Example 3: 4X - 5 17X

Let's solve 4X - 5 17X.

Move all terms with X to one side and all constants to the other: 4X - 17X 5 Combine like terms: -13X 5 Divide both sides by -13 to isolate X: -13X ÷ -13 5 ÷ -13 Simplify to: X -5/13 Verify by substituting X -5/13 into the original equation: 4(-5/13) - 5 17(-5/13) -20/13 - 5 -85/13 -20/13 - 65/13 -85/13 -85/13 -85/13

This confirms the solution is correct.

Important Tips for Solving Linear Equations

When solving linear equations, always remember these crucial points:

Balance is Key: Whatever operation you perform on one side of the equation, you must perform it on the other side to maintain equality. Isolate the Variable: The goal is to get the variable alone on one side of the equation. Verify Your Solution: Always check your solution by substituting it back into the original equation.

By following these guidelines, you can confidently solve any linear equation and ensure the accuracy of your solution.