Linear Equations and Lines
Quick Reference
| Textbook | OpenStax College Algebra 2e |
| Chapter | Chapter 2: Equations and Inequalities |
| Section | 2.2 Linear Equations in One Variable |
| Subsection | Slope and lines support in 2.1 (The Rectangular Coordinate Systems and Graphs) |
| Pages | Page numbers pending faculty verification. |
| Course | MATH141 (Precalculus I) |
Try This First
Before any formula, work this small case with paper or in your head.
Two friends record how far they have walked along a straight trail at two moments. After 2 minutes the first is at the 100 meter marker. After 6 minutes the same person is at the 340 meter marker.
- How many meters did the walker cover between those two moments?
- How many minutes passed?
- If the pace stayed steady, how many meters were covered each minute?
Reveal one way to think about it
Distance covered is $340 - 100 = 240$ meters. Time passed is $6 - 2 = 4$ minutes. Meters per minute is $\frac{240}{4} = 60$. The walker moved at a steady 60 meters each minute.
That single number, 60 meters per minute, is the rate at which distance grows as time grows. When the same idea is written for a graph, it is called the slope of the line. Slope is one move: change in output divided by change in input.
Quantity First: What Changes With What
A line records a relationship where the output changes by the same amount for every equal step in the input. Walk one more minute, gain 60 more meters. Walk one more minute, gain 60 more meters again. The amount of change per step never shifts.
That steadiness is what defines a linear relationship. Compare it to a relationship that speeds up: a ripple on a pond widens slowly at first and faster later, so equal steps in time do not give equal steps in radius. A line is the case where equal input steps always produce equal output steps.
Two representations of the same steady walk appear below. Read across the table, then check the rule, and notice they describe the same motion.
| Time $t$ (min) | Distance $d$ (m) |
|---|---|
| 0 | $-20$ |
| 2 | 100 |
| 4 | 220 |
| 6 | 340 |
Algebraic rule for this table: $d = 60t - 20$.
Each two-minute step adds 120 meters in the table, which is 60 meters per minute, matching the number multiplied by $t$ in the rule. Translating between the table and the rule is the skill to build here: the per-step change in the table is the coefficient on the input in the rule.
Prerequisite Hub
How this skill connects to its prerequisites and what it leads to.
Builds On
| Skill | Why it is needed |
|---|---|
Algebraic Simplification (algebraic-simplification) |
Combining like terms and distributing correctly keeps each solving step honest |
Solving Basic Equations (solving-equations-basic) |
Undoing operations in reverse order is the engine of every linear solve |
Unlocks
| Skill | What it opens |
|---|---|
Absolute Value Equations (absolute-value) |
Splits into two linear cases, each solved by the method here |
Quadratic Equations (quadratic-equations) |
Builds on the linear solve once a squared term appears |
Solving Inequalities (solving-inequalities) |
Same balancing moves, with a sign-flip rule when multiplying by a negative |
Rates of Change (m141-rates-of-change) |
Slope becomes average rate of change, the bridge into calculus |
Cross-Course Prerequisites
None recorded for this node.
The Official Definition
Slope of a line. The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is
$$\boxed{m = \frac{y_2 - y_1}{x_2 - x_1}}$$
the change in output (rise) divided by the change in input (run).
Source: OpenStax College Algebra 2e, Section 2.2 Linear Equations in One Variable (slope-intercept form $y = mx + b$).
A few readings of this one formula keep it from feeling like a memorized string.
- The numerator $y_2 - y_1$ is how much the output changed between the two points.
- The denominator $x_2 - x_1$ is how much the input changed between the same two points.
- Their ratio is how much output you gain for each one unit of input, which is exactly the steady per-step change from the table above.
Slope-Intercept Form
A line can be written as
$$y = mx + b$$
where $m$ is the slope (the steady per-step change in output) and $b$ is the $y$-intercept (the output when the input is $0$). In the walking table, $m = 60$ and $b = -20$, so $d = 60t - 20$.
Linear Equation in One Variable
A linear equation in one variable is an equation that can be written in the form $ax + b = 0$ with $a \neq 0$. Solving it means finding the single input value that makes the statement true. The balancing principle does the work: whatever operation is applied to one side is applied to the other, which keeps the equality true while simplifying its form.
Worked Examples
Each example states a prediction before computing, then checks the result against that prediction.
Example 1: Slope From Two Points
Find the slope of the line through $(1, 2)$ and $(4, 11)$.
Predict. The output climbs from 2 to 11 (an increase) while the input climbs from 1 to 4. A rising output with a rising input gives a positive slope, and the output climbs faster than the input, so expect a slope above 1.
Compute. Take $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 11)$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3$$
Check. The slope is $3$, which is positive and above 1, matching the prediction. Each step of 1 in the input raises the output by 3.
Example 2: The Order Does Not Matter, But Be Consistent
Find the slope through $(4, 11)$ and $(1, 2)$, reversing the order from Example 1.
Predict. The geometric line is identical, so the slope must be the same value, $3$.
Compute. Now $(x_1, y_1) = (4, 11)$ and $(x_2, y_2) = (1, 2)$.
$$m = \frac{2 - 11}{1 - 4} = \frac{-9}{-3} = 3$$
Check. The slope is again $3$. Subtracting in the opposite order flips the sign of both the numerator and the denominator, and the two sign flips cancel. The rule that matters: use the same point first in both the top and the bottom.
Example 3: Solving a Linear Equation in One Variable
Solve $3(x - 2) + 4 = 2x + 5$.
Predict. The left side grows like $3x$ and the right like $2x$, so one input value should balance them. The constant on the right is modest, so expect a small whole number or simple value.
Compute.
$$3(x - 2) + 4 = 2x + 5$$
Distribute on the left.
$$3x - 6 + 4 = 2x + 5$$
Combine like terms on the left.
$$3x - 2 = 2x + 5$$
Subtract $2x$ from both sides (the same operation on each side keeps the balance).
$$x - 2 = 5$$
Add $2$ to both sides.
$$x = 7$$
Check. Substitute $x = 7$ into the original equation. Left side: $3(7 - 2) + 4 = 3(5) + 4 = 19$. Right side: $2(7) + 5 = 19$. Both sides equal $19$, so $x = 7$ is correct, and it is a small whole number as predicted.
Example 4: Writing the Equation of a Line
Write the slope-intercept equation of the line through $(2, 100)$ with slope $60$.
Predict. With slope $60$ and a point well above the origin, the $y$-intercept $b$ should be a smaller number than 100, because moving back from input $2$ to input $0$ subtracts two steps of $60$.
Compute. Start from $y = mx + b$ with $m = 60$, then use the point to find $b$.
$$100 = 60(2) + b$$
$$100 = 120 + b$$
$$b = 100 - 120 = -20$$
So the line is $y = 60x - 20$.
Check. Stepping back from input $2$ to input $0$ removes $2 \times 60 = 120$ from the output, giving $100 - 120 = -20$, which matches $b$ and matches the walking table from earlier in this lesson.
Common Misconceptions
a steep climb on the graph means a large output value. Slope and height are different quantities. Slope measures how fast the output is changing; height measures the output itself. A line can sit high above the axis while staying perfectly flat (slope $0$), and a line can pass through low output values while climbing steeply. Consider $y = 1000$ (a high, flat line with slope $0$) against $y = 5x$ near $x = 1$ (a low output of $5$ but a steep slope of $5$). Reading steepness off the height confuses the rate of change with the value, which is the height-vs-slope error.
Predict-then-check. Before reading the resolution below, predict: for the line $y = 1000$, what is the slope?
Check your prediction
The slope is $0$. Pick any two points, such as $(0, 1000)$ and $(7, 1000)$. The output never changes, so $y_2 - y_1 = 1000 - 1000 = 0$, and $m = \frac{0}{7 - 0} = 0$. The line sits at a large height yet has zero slope, which separates “how high” from “how fast it changes.”
subtracting the coordinates in inconsistent order changes the slope. The slope formula allows either point to be first, but the same point must lead in both the numerator and the denominator. Writing $\frac{y_2 - y_1}{x_1 - x_2}$ mixes the orders and flips only the denominator’s sign, producing the negative of the true slope. The fix is to keep the subtraction order identical top and bottom.
Predict-then-check. For the points $(1, 2)$ and $(4, 11)$, predict the value of the mixed-up expression $\frac{11 - 2}{1 - 4}$, then compare it to the correct slope of $3$.
Check your prediction
The mixed expression gives $\frac{9}{-3} = -3$, the negative of the correct slope. The numerator used the order $y_2 - y_1$ while the denominator used $x_1 - x_2$, so only one part changed sign. Keeping both in the order $(\text{second}) - (\text{first})$ restores $\frac{9}{3} = 3$.
Leveled Practice
Work each problem before opening its answer. A wrong turn here is common and shows something worth understanding; there is no need to rush.
Level 1: Read a Rate From a Table
A candle is 20 cm tall and burns down 3 cm every hour. Fill in the table and state how the height changes per hour.
| Hours burning | Height (cm) |
|---|---|
| 0 | ? |
| 1 | ? |
| 2 | ? |
Show answer
| Hours burning | Height (cm) |
|---|---|
| 0 | 20 |
| 1 | 17 |
| 2 | 14 |
The height decreases by $3$ cm each hour, so the per-hour change is $-3$ cm per hour.
Level 2: Slope From Two Points
Find the slope of the line through $(-2, 5)$ and $(3, -5)$.
Show answer
$$m = \frac{-5 - 5}{3 - (-2)} = \frac{-10}{5} = -2$$
The slope is $-2$. The output falls by $2$ for every $1$ unit the input rises.
Level 3: Solve a Linear Equation
Solve $5x - 3 = 2x + 12$.
Show answer
Subtract $2x$ from both sides: $3x - 3 = 12$. Add $3$ to both sides: $3x = 15$. Divide both sides by $3$: $x = 5$.
Check: left side $5(5) - 3 = 22$; right side $2(5) + 12 = 22$. Both equal $22$, so $x = 5$.
Level 4: Build the Equation of a Line
Write the slope-intercept equation of the line through $(0, 4)$ and $(3, 13)$.
Show answer
Slope first: $m = \frac{13 - 4}{3 - 0} = \frac{9}{3} = 3$. The point $(0, 4)$ has input $0$, so its output is the $y$-intercept: $b = 4$.
The line is $y = 3x + 4$.
Check with the second point: $3(3) + 4 = 13$, which matches $(3, 13)$.
Level 5: Reason About a Constraint
A line passes through $(1, 7)$ and has the same slope as the line $y = -4x + 9$. Find its equation, then explain in one sentence why two different lines can share a slope.
Show answer
The slope of $y = -4x + 9$ is $-4$, so the new line also has slope $-4$. Start from $y = -4x + b$ and use the point $(1, 7)$:
$$7 = -4(1) + b \implies 7 = -4 + b \implies b = 11$$
The line is $y = -4x + 11$.
Two different lines can share a slope because slope fixes only the steepness and direction of the climb, not the height where the line sits; the intercept $b$ shifts the whole line up or down without changing how fast it rises or falls. These lines are parallel.
Why Might This Be True
Pick one of these to think through, or to explain to a classmate.
- The slope formula divides change in output by change in input. Why must the denominator $x_2 - x_1$ be nonzero, and what does it mean for a “line” when the two points share the same input? (One way to see it: a vertical line has the same input for many outputs, so the run is $0$ and the slope is undefined.)
- One way to write a line is slope-intercept form $y = mx + b$. Another way is to start from a single point and the slope. Which do you reach for first, and why does it depend on what information a problem hands you?
Tie it back to meaning: in the candle problem from Level 1, the slope was $-3$. State in one plain sentence what that $-3$ tells someone watching the candle, with no algebra symbols.
Mastery Checklist
Novice (Level 1-2):
Competent (Level 3-4):
Proficient (Level 5):
Connections
Looking back:
- Algebraic Simplification supplies the combining and distributing each solving step relies on
- Solving Basic Equations supplies the inverse-operation moves that isolate the variable
Looking ahead:
- Absolute Value Equations split into two linear cases, each handled by the method here
- Quadratic Equations extend the solving toolkit once a squared term appears
- Solving Inequalities reuse the same balancing moves with one extra sign rule
- Rates of Change reinterprets slope as average rate of change, the on-ramp to the derivative
Real-world connections:
- A steady wage builds total pay as a line in hours worked, with hourly pay as the slope
- A constant download speed grows a file as a line in seconds, with the speed as the slope
- Any quantity that changes by a fixed amount per equal step traces a line
Resources
- Primary textbook section: OpenStax College Algebra 2e, Section 2.2 Linear Equations in One Variable. https://openstax.org/books/college-algebra-2e/pages/2-2-linear-equations-in-one-variable
- Coordinate and graph support: OpenStax College Algebra 2e, Section 2.1 The Rectangular Coordinate Systems and Graphs. https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs
Both sources are free and openly licensed.
| Previous | Up | Next |
|---|---|---|
| Solving Basic Equations | Skills Index | Solving Inequalities |
Last updated: 2026-06-16