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Linear equations are an important part of algebra and can appear in placement-test preparation. Students preparing for ACCUPLACER math should understand how to isolate a variable, maintain equality, and translate written problems into equations. The good news is that many linear equations follow the same basic process. Once you understand the rules, you can apply them to a wide range of questions.
What is a linear equation?
A linear equation is an equation in which the variable has a power of one. A simple example is: \[ 2x+5=17 \] The goal is to find the value of \(x\) that makes the equation true. To solve it, use inverse operations to isolate the variable.
Step 1: Remove addition or subtraction
Start with: \[ 2x+5=17 \] Subtract five from both sides: \[ 2x=12 \] You must perform the same operation on both sides to keep the equation balanced.
Step 2: Remove multiplication or division
The variable is currently multiplied by two. Divide both sides by two: \[ x=6 \] The solution is six. You can think of solving an equation as reversing the operations applied to the variable. Addition is reversed by subtraction, and multiplication is reversed by division.
Step 3: Check your answer
Substitute six into the original equation: \[ 2(6)+5=17 \] \[ 12+5=17 \] Both sides equal 17, so the answer is correct. Checking your work is especially useful when you are learning a new method or solving questions with several steps.
Equations with variables on both sides
Consider: \[ 5x+2=3x+10 \] Subtract \(3x\) from both sides: \[ 2x+2=10 \] Subtract two: \[ 2x=8 \] Divide by two: \[ x=4 \] The same principle applies: perform matching operations on both sides until the variable is isolated.
Solving equations with parentheses
You could also use the distributive property first. Both approaches are valid when applied correctly.
Translate word problems into equations
Algebra becomes more useful when you can turn a written situation into a mathematical expression. Imagine that a service charges a fixed fee of $8 plus $3 for each item. If the total is $26, how many items were purchased? Let \(x\) represent the number of items
The customer purchased six items. When solving word problems, define the variable, identify the known quantities, and decide which mathematical relationship connects them.
Common mistakes to avoid
Students sometimes change only one side of an equation, make sign errors when moving terms, or forget to distribute a number across every term inside parentheses. Rather than relying on the phrase “move it to the other side,” think about the actual operation being performed. This makes it easier to understand why the sign changes.
Build your ACCUPLACER skills
Practice equations that involve one operation, several operations, parentheses, and variables on both sides. Once you become comfortable with the process, move to word problems.
