Solving Linear Equations: A Step-by-Step Guide

Linear equations are fundamental in algebra and are widely used in various fields like physics, engineering, and economics. Solving linear equations involves finding the value of the unknown variable that makes the equation true. This guide provides a step-by-step approach to solving linear equations for x, covering the essential rules of algebra and the order of operations.

Understanding Linear Equations

A linear equation is an equation where the highest power of the variable is 1. It typically takes the form ax + b = c, where a, b, and c are constants, and x is the variable. The goal is to isolate x on one side of the equation to find its value.

Steps to Solve Linear Equations

Here's a comprehensive guide to solving linear equations:

1. Simplify both sides of the equation:
   • Combine like terms on each side of the equation.
   • Distribute any coefficients if necessary.
2. Isolate the variable term:
   • Use the addition or subtraction property of equality to move all terms containing the variable to one side of the equation.
   • Use the multiplication or division property of equality to get the variable term by itself.
3. Solve for the variable:
   • Divide both sides of the equation by the coefficient of the variable to isolate the variable.
4. Check your solution:
   • Substitute the solution back into the original equation to verify that it makes the equation true.

Examples

Example 1:

Solve the equation 2x + 5 = 11.

1. Simplify both sides: The equation is already simplified.
2. Isolate the variable term: Subtract 5 from both sides: 2x = 6
3. Solve for the variable: Divide both sides by 2: x = 3
4. Check the solution: Substitute x = 3 into the original equation: 2(3) + 5 = 11. This simplifies to 6 + 5 = 11, which is true. Therefore, x = 3 is the correct solution.

Example 2:

Solve the equation 3(x - 2) = 9.

1. Simplify both sides: Distribute the 3 on the left side: 3x - 6 = 9
2. Isolate the variable term: Add 6 to both sides: 3x = 15
3. Solve for the variable: Divide both sides by 3: x = 5
4. Check the solution: Substitute x = 5 into the original equation: 3(5 - 2) = 9. This simplifies to 3(3) = 9, which is true. Therefore, x = 5 is the correct solution.

Conclusion

Solving linear equations is a fundamental skill in algebra. By following the step-by-step guide outlined above, you can confidently solve various linear equations. Remember to simplify both sides, isolate the variable term, and always check your solution to ensure accuracy.