5.2 Linear Equations in One Variable

Linear Equations in One Variable

Quick Answer: A linear equation in one variable can be written as ax + b = c, where a != 0. On the Florida Algebra 1 EOC, MA.912.AR.2.1 tests your ability to write a one-variable linear equation from a real-world context and solve multi-step equations with variables on both sides and rational coefficients, while MA.912.AR.2.6 tests solving and graphing one-variable linear inequalities.

Anatomy of a Linear Equation

A linear equation in one variable has a single variable raised only to the first power. The solution is the value of the variable that makes the equation true. Standard form: ax + b = c (or ax + b = cx + d when the variable appears on both sides).

Solving Strategy: Undo in Reverse Order of Operations

The reliable strategy is to isolate the variable by performing inverse operations in reverse order:

1. Clear parentheses using the distributive property.
2. Clear fractions by multiplying both sides by the least common denominator (LCD).
3. Combine like terms on each side.
4. Move variable terms to one side.
5. Move constants to the other side.
6. Divide by the coefficient of the variable.

Worked example (rational coefficients): (1/2)x + (2/3) = (5/6)x.

• LCD of 2, 3, 6 is 6. Multiply each term by 6: 3x + 4 = 5x. Subtract 3x: 4 = 2x. Divide by 2: x = 2.

Worked example (variables on both sides with distribution): 3(x - 4) = 2x + 5 - x.

• Distribute and combine on the right: 3x - 12 = x + 5.
• Subtract x: 2x - 12 = 5. Add 12: 2x = 17. Divide by 2: x = 8.5.

Writing Equations from Context

Florida items commonly present a word problem and ask students to write the equation before solving it. Translate phrases carefully:

Worked example: A rental company charges a flat fee of $25 plus $0.40 per mile. A customer's total bill was $73. Write and solve an equation for miles driven.

• Equation: 25 + 0.40m = 73. Subtract 25: 0.40m = 48. Divide by 0.40: m = 120 miles.

Types of Solutions: One, None, Infinite

When the variable cancels out entirely, the equation has either no solution (contradiction) or infinitely many solutions (identity). Florida EOC items use these special cases as distractors.

• One solution: The variable has a unique value. Example: 2x + 3 = 11 gives x = 4.
• No solution: The equation reduces to a false statement. Example: 3(x + 2) = 3x + 7 simplifies to 6 = 7. Solution set is empty.
• Infinite solutions: The equation reduces to a true statement. Example: 4x + 8 = 2(2x + 4) simplifies to 8 = 8. Any real value of x works.

Exam trap: If your variable terms cancel and you get a numerical statement, do not write "x = 0." A true statement means infinite solutions; a false statement means no solution.

One-Variable Linear Inequalities

Solving a linear inequality follows the same procedure as solving an equation with one critical exception: when you multiply or divide both sides by a negative number, reverse the inequality symbol.

Worked example: -3(x - 2) >= 9 - 3x.

• Distribute: -3x + 6 >= 9 - 3x. Add 3x: 6 >= 9. This is false, so the inequality has no solution.

Graphing on a number line: Use an open circle for < or > (endpoint not included) and a closed circle for <= or >= (endpoint included). Shade the direction of the solution set.

Common Exam Traps

1. Forgetting to flip the inequality sign when dividing by a negative. This is the single most tested inequality error.
2. Distributing a negative coefficient incorrectly: -2(x - 5) = -2x + 10, not -2x - 10.
3. Combining unlike terms when clearing parentheses: 3(x + 4) = 3x + 12, not 3x + 4.
4. Treating "no solution" as x = 0. Verify by substituting back into the original equation.
5. Misreading the question: "Write an equation" items ask for the equation itself, not the solution.

Real-World Inequality Context

Florida items often phrase inequalities as minimum or maximum thresholds. Example: A student needs at least 75 points on a final test to pass. The test has 20 questions worth 5 points each, and the student already earned 35 points. Let x = number of correct answers on the final:

• Inequality: 5x + 35 >= 75. Subtract 35: 5x >= 40. Divide: x >= 8.

This pattern applies to minimum GPA requirements, spending limits, and speed thresholds — common EOC items.