TSI Math Study Guide
Linear Equations and Functions
Linear equations describe relationships in which one quantity changes at a constant rate compared with another.
Solving linear equations
Isolate the variable by performing the same operation on both sides.
Check it
Substitute back: 5(6) − 7 = 30 − 7 = 23. True, so x = 6 is correct.
Variables on both sides
Move the variable terms to one side and the constants to the other.
Equations with parentheses
Distribute before combining like terms.
The −2 multiplies both terms inside the parentheses.
Linear inequalities
Solve an inequality the same way as an equation.
When you multiply or divide both sides by a negative number, reverse the inequality sign.
The solution includes −5 (closed circle) and every number greater than −5.
Understanding slope
m = y2 − y1x2 − x1 = change in ychange in x = riserun
Positive slope
Rises from left to right.
Negative slope
Falls from left to right.
Zero slope
Horizontal line.
Slope-intercept form
Slope 3, y-intercept 7. The line crosses the y-axis at (0, 7). From there, a slope of 3 = 31 means the line rises 3 units for every 1 unit to the right.
Writing the equation of a line
From slope and intercept
Slope −2, y-intercept 5. Substitute directly:y = −2x + 5
From two points
Through (2, 7) and (5, 16).
Linear relationships in tables
A table is linear when the rate of change is constant.
| x | y | change in y |
|---|---|---|
| 0 | 4 | — |
| 1 | 7 | +3 |
| 2 | 10 | +3 |
| 3 | 13 | +3 |
Read both values off the table
y rises 3 each time x rises 1, so m = 3. When x = 0, y = 4, so b = 4.y = 3x + 4
Tables give you both the rate of change and the starting value directly.
Interpreting slope and intercept in context
Slope = 8
The rental costs $8 per hour.
Intercept = 12
There’s a $12 initial charge before any hourly charges.
In real-world problems
Slope = rate of change · Intercept = starting value
The units tell you how to interpret each number.
Functions
f(x) does not mean f multiplied by x. It identifies the output produced by a particular input.
Evaluating a function
If f(x) = 4x − 3, find f(5).
Finding an input from an output
If f(x) = 3x + 4, when is f(x) = 19?
Connecting equations, tables, and graphs
A linear relationship may appear as an equation, a table, a graph, or a word problem — different views of the same information.
Equation
y = 5x + 2Slope 5, intercept 2. y rises 5 whenever x rises 1.
Table
| x | y |
|---|---|
| 0 | 2 |
| 1 | 7 |
| 2 | 12 |
| 3 | 17 |
Graph
Recognizing these connections is more useful than treating them as separate topics.
TSI strategy: look for the rate and starting value
How much does the output change when the input increases by 1?
That’s the slope.
What is the output when the input is zero?
That’s the y-intercept.
This works whether the information comes from a table, graph, equation, or word problem. For equations, remember that checking is quick: substitute your answer back into the original and see if it’s true.
Don’t memorize y = mx + b without understanding what m and b represent.
Slope = rate of changey-intercept = starting value
That interpretation makes it much easier to move among equations, tables, graphs, and word problems.