3.1 Linear equations & functions

Linear Equations and Functions

A linear equation is an equation in which the variable appears only to the first power — there are no exponents, square roots, or variables in a denominator. On the TSIA2 Mathematics test you will solve linear equations, graph and interpret linear functions, and work with systems of two linear equations. The inverse-operation routine you practice here shows up throughout the whole exam, so make it automatic.

Solving Linear Equations

To solve a linear equation, undo the operations in reverse order until the variable stands alone. Whatever you do to one side, you must do to the other.

Example 1. Solve 3x + 7 = 22. Subtract 7 from both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3(5) + 7 = 15 + 7 = 22. Correct.

Example 2 (variable on both sides). Solve 5x − 4 = 2x + 11. Subtract 2x from both sides: 3x − 4 = 11. Add 4 to both sides: 3x = 15. Divide by 3: x = 5. Check: 5(5) − 4 = 21 and 2(5) + 11 = 21. Both sides equal 21, so it works.

Example 3 (distribute first). Solve 2(x + 3) = 4x − 2. Distribute the 2: 2x + 6 = 4x − 2. Subtract 2x: 6 = 2x − 2. Add 2: 8 = 2x, so x = 4. Check: 2(4 + 3) = 14 and 4(4) − 2 = 14. Correct.

Slope-Intercept Form

The equation y = mx + b is called slope-intercept form. Here m is the slope (the steepness of the line) and b is the y-intercept (the y-value where the line crosses the y-axis). Slope is the ratio of vertical change to horizontal change between any two points:

slope m = rise / run = (y₂ − y₁) / (x₂ − x₁).

Example 4. Find the equation of the line through (1, 2) and (4, 11). Slope: m = (11 − 2) / (4 − 1) = 9 / 3 = 3. Now find b using y = 3x + b with the point (1, 2): 2 = 3(1) + b, so b = −1. Equation: y = 3x − 1.

Point-Slope Form

When you know the slope and one point, the fastest tool is point-slope form: y − y₁ = m(x − x₁).

Example 5. Write the line with slope 2 that passes through (3, 5). y − 5 = 2(x − 3) y − 5 = 2x − 6 y = 2x − 1.

Function Notation

A function pairs each input with exactly one output. We write f(x), read "f of x," for the output when the input is x. The parentheses do not mean multiplication — they hold the input.

Example 6. If f(x) = 2x − 1, then:

• Evaluate f(4): f(4) = 2(4) − 1 = 8 − 1 = 7.
• Solve f(x) = 9: set 2x − 1 = 9, so 2x = 10 and x = 5.

Domain and Range

The domain is the set of allowed inputs (x-values), and the range is the set of possible outputs (y-values). For a non-constant linear function such as f(x) = 2x − 1, the domain and the range are both all real numbers. Restrictions appear only when a real situation limits the inputs — for instance, if x counts hours worked, then the domain is x ≥ 0 because negative hours make no sense.

Systems of Linear Equations

A system is two equations solved together; the solution is the single (x, y) point that satisfies both equations at once.

Substitution — Example 7. y = 2x + 1 3x + y = 11 Replace y in the second equation with 2x + 1: 3x + (2x + 1) = 11, so 5x + 1 = 11, giving 5x = 10 and x = 2. Then y = 2(2) + 1 = 5. Solution: (2, 5).

Elimination — Example 8. 2x + 3y = 12 2x − y = 4 Subtract the second equation from the first so the x-terms cancel: (3y − (−y)) = 12 − 4, which gives 4y = 8, so y = 2. Substitute back: 2x − 2 = 4, so 2x = 6 and x = 3. Solution: (3, 2).

Interpreting Linear Functions

In a real context the slope is a rate of change and the y-intercept is a starting value. Suppose a phone plan costs C = 0.10m + 20, where m is the number of minutes used.

Reading the slope and intercept from a formula, table, or graph is one of the most common TSIA2 question types. Always ask two questions: "What changes per step?" (that is the slope) and "Where does it start?" (that is the intercept). Answer those and you can interpret almost any linear model on the test.