What is a linear equation?
A linear equation is an equation in which the variable appears to the first power. For example, \(2x+5=17\) is a linear equation because the variable \(x\) is not squared or raised to another power. Linear equations are useful for representing unknown quantities in everyday situations. ACCUPLACER math questions may ask you to solve equations directly or use them to answer word problems.
1. Understand the goal
The main goal is to find the value of the variable that makes both sides of the equation equal. Consider: \[ x+7=15 \] Subtract 7 from both sides: \[ x+7-7=15-7 \] Therefore, \(x=8\). The equation stays balanced because the same operation was performed on both sides.
2. Use inverse operations
Inverse operations undo one another. - Addition and subtraction are inverses. - Multiplication and division are inverses. For example: \[ 4x=28 \] Divide both sides by 4: \[ x=7 \] When an equation contains several operations, reverse them in an appropriate order to isolate the variable.
3. Solve two-step equations
Consider the equation: \[ 3x+4=19 \] First subtract 4 from both sides: \[ 3x=15 \] Then divide both sides by 3: \[ x=5 \] A common mistake is to divide by 3 before removing the added 4. Follow the order of operations in reverse when isolating the variable.
4. Work with variables on both sides
Some equations have variables on both sides: \[ 5x+2=2x+14 \] Subtract \(2x\) from both sides: \[ 3x+2=14 \] Subtract 2: \[ 3x=12 \] Divide by 3 to get \(x=4\). If the variable terms cancel and the remaining statement is true, the equation may have infinitely many solutions. If they cancel and leave a false statement, there is no solution.
5. Translate word problems into equations
Suppose a gym charges a $10 registration fee and $15 per month. You want to calculate the total cost after four months. The equation is: \[ C=10+15m \] Substituting \(m=4\): \[ C=10+15(4)=70 \] The total is $70. Identify what the unknown represents, determine the relationship between the quantities, and write the equation before calculating.
6. Check your answer
Substitute the proposed answer into the original equation. For \(3x+4=19\), the answer \(x=5\) gives \(3(5)+4=19\), so it checks out.
Final tip:
Practice simple equations until the steps feel natural. Then move to equations with parentheses, fractions, and variables on both sides. Always check signs carefully and verify your answer when time permits.
