Solving Linear Equations: A Step-by-Step Guide

Linear equations are mathematical expressions that represent a straight line when graphed. They are of the form:

$ax + b = c$

where:

• $a$, $b$, and $c$ are constants (numbers),
• $x$ is the variable we need to solve for.

Steps to Solve a Linear Equation

1. Simplify Both Sides

• If the equation has parentheses, distribute to eliminate them.
• Combine like terms on each side of the equation.

2. Isolate the Variable

• To get the variable ($x$) on one side, we need to "undo" the operations around it by performing opposite operations (addition becomes subtraction, multiplication becomes division, etc.).

• Start by adding or subtracting to move any constants ($b$ or $c$) to the other side.

$ax + b = c \quad \Rightarrow \quad ax = c - b$

3. Solve for the Variable

• Once the variable term ($ax$) is isolated, solve for $x$ by dividing by the coefficient of $x$ (which is $a$).

$ax = d \quad \Rightarrow \quad x = \frac{d}{a}$

4. Check Your Solution

• Substitute the value of $x$ back into the original equation to verify that both sides are equal.

Example Problem

Solve the equation: $3x + 5 = 14$.

Step 1: Simplify Both Sides

• The equation is already simplified.

Step 2: Isolate the Variable

• Subtract 5 from both sides:

$3x + 5 - 5 = 14 - 5$ $3x = 9$

Step 3: Solve for the Variable

• Divide both sides by 3:

$x = \frac{9}{3} = 3$

Step 4: Check Your Solution

• Substitute $x = 3$ back into the original equation:

$3(3) + 5 = 14 \quad \text{(True!)}$

Summary

• Simplify both sides of the equation.
• Isolate the variable by moving terms around.
• Solve for the variable.
• Check your solution.

Do you want more details on any of these steps, or do you have any questions?

Related Questions:

1. How do you solve a linear equation with variables on both sides?
2. What happens if a linear equation has no solution?
3. How can you solve a linear equation with fractions?
4. How do you solve a linear equation with decimals?
5. What are the properties of equality used in solving equations?

Tip: Always perform the same operation on both sides of the equation to maintain the equality!