Writing Linear Equations

In the previous lesson, you learned how to read a linear equation and understand what it tells you about a line. Now you are going to reverse the process. Instead of starting with an equation and figuring out the graph, you will start with information about a line and write the equation yourself. Think of it as reverse engineering. You already have all the tools you need. This lesson is about putting them together.

Core Concepts

Starting with Slope and Y-Intercept

The simplest case is when someone hands you the slope and the y-intercept directly. If you know both of these, you can write the equation immediately using slope-intercept form:

$$y = mx + b$$

where $m$ is the slope and $b$ is the y-intercept.

For example, if someone tells you “the slope is 4 and the y-intercept is $-3$,” you just plug those values in: $y = 4x - 3$. That is the equation. Done.

This feels almost too easy, and that is because it is the ideal situation. In real life, you rarely get handed exactly the information you need in exactly the form you want it. More often, you have to work a bit to extract the slope and y-intercept from other clues.

Writing Equations Given Slope and a Point

What if you know the slope and one point on the line, but that point is not the y-intercept? This is where point-slope form becomes useful.

Point-slope form is:

$$y - y_1 = m(x - x_1)$$

where $m$ is the slope and $(x_1, y_1)$ is any point on the line.

Here is the intuition: the slope tells you the relationship between any two points on the line. If you know one point and the slope, you can describe every other point on the line by saying “start at $(x_1, y_1)$ and use the slope to find the change.”

The formula $y - y_1 = m(x - x_1)$ is really just the slope formula rearranged:

$$m = \frac{y - y_1}{x - x_1}$$

Multiply both sides by $(x - x_1)$ and you get point-slope form.

Once you write the equation in point-slope form, you can distribute and simplify to get slope-intercept form if that is what you need. Both forms describe the same line.

Writing Equations Given Two Points

If you have two points but no slope, no problem. You can find the slope yourself using the slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Once you have the slope, pick either of your two points and use point-slope form to write the equation.

The process is:

1. Use the two points to calculate the slope
2. Plug the slope and one of the points into point-slope form
3. Simplify to slope-intercept form if desired

It does not matter which point you choose for point-slope form. Both will give you the same final equation because both points are on the same line.

Writing Equations from Graphs

When you have a graph, you are looking at a visual representation of all the information you need. Your job is to extract two key pieces:

1. Find the slope. Pick two points where the line crosses grid intersections (so you can read the coordinates clearly). Then count the rise and run between them, or use the slope formula.

2. Find the y-intercept. Look for where the line crosses the y-axis. The y-coordinate at that point is your value of $b$.

Once you have both, plug them into $y = mx + b$.

If the y-intercept is not easy to read from the graph (maybe it falls between grid lines), just use any two clear points and the point-slope method instead.

Writing Equations from Real-World Scenarios

Word problems often describe linear relationships without using mathematical notation. Your job is to translate the English into an equation.

Look for these clues:

• Rate of change (per hour, per mile, per item) usually becomes the slope
• Starting value or initial amount usually becomes the y-intercept
• Total amount is typically what you are solving for (the dependent variable, $y$)

For example: “A plumber charges $75 for a service call plus $50 per hour of labor.”

• The $50 per hour is the rate of change (slope): $m = 50$
• The $75 is the starting charge (y-intercept): $b = 75$
• The equation is $y = 50x + 75$, where $x$ is hours worked and $y$ is total cost

Parallel Lines

Two lines are parallel if they never intersect. What would cause two different lines to never cross? They would have to be tilting at exactly the same angle, going in exactly the same direction. That means parallel lines have the same slope.

If you need to write an equation parallel to a given line:

1. Find the slope of the given line
2. Use that same slope for your new line
3. Use point-slope form with the new point you are given

For example, if you need a line parallel to $y = 2x + 5$ that passes through $(3, 1)$:

• The slope of the given line is 2
• Your parallel line also has slope 2
• Use point-slope form: $y - 1 = 2(x - 3)$
• Simplify: $y = 2x - 5$

The two equations have the same slope (2) but different y-intercepts (5 versus $-5$), so they are parallel.

Perpendicular Lines

Two lines are perpendicular if they meet at a 90-degree angle, forming an L-shape or plus sign. For this to happen, the slopes must have a very specific relationship: they must be negative reciprocals of each other.

To find the negative reciprocal of a number:

1. Take the reciprocal (flip the fraction)
2. Change the sign (positive becomes negative, negative becomes positive)

For example:

• The negative reciprocal of 2 is $-\frac{1}{2}$
• The negative reciprocal of $-3$ is $\frac{1}{3}$
• The negative reciprocal of $\frac{2}{5}$ is $-\frac{5}{2}$

Here is why this works: when you multiply a number by its negative reciprocal, you always get $-1$. Try it: $2 \times (-\frac{1}{2}) = -1$. This mathematical relationship is what creates the 90-degree angle.

If you need to write an equation perpendicular to a given line:

1. Find the slope of the given line
2. Find the negative reciprocal of that slope
3. Use point-slope form with the new slope and the point you are given

Notation and Terminology

Term	Meaning	Example
Point-slope form	$y - y_1 = m(x - x_1)$	$y - 3 = 2(x - 1)$
Parallel lines	Lines with same slope, never intersect	$y = 2x + 1$ and $y = 2x - 3$
Perpendicular lines	Lines meeting at 90 degrees, slopes are negative reciprocals	$y = 2x$ and $y = -\frac{1}{2}x$
Negative reciprocal	Flip and change sign	Negative reciprocal of 2 is $-\frac{1}{2}$

Examples

Key Properties and Rules

Forms of Linear Equations

Slope-intercept form: $y = mx + b$

• Use when you know slope and y-intercept
• Best for graphing and quick analysis

Point-slope form: $y - y_1 = m(x - x_1)$

• Use when you know slope and any point
• Best for writing equations from given information

Standard form: $Ax + By = C$

• Useful for certain algebraic operations
• Convert to slope-intercept form to find slope: $m = -\frac{A}{B}$

Strategy Summary

Given Information	Strategy
Slope and y-intercept	Plug directly into $y = mx + b$
Slope and a point	Use point-slope form, then simplify
Two points	Calculate slope first, then use point-slope form
Graph	Read slope (rise/run) and y-intercept, use $y = mx + b$
Parallel to a line	Use same slope as given line
Perpendicular to a line	Use negative reciprocal of given slope

Parallel and Perpendicular Slopes

Parallel lines: Same slope, different y-intercepts

If Line 1 has slope $m$, then any parallel line also has slope $m$.

Perpendicular lines: Negative reciprocal slopes

If Line 1 has slope $m$, then any perpendicular line has slope $-\frac{1}{m}$.

Quick check: Two lines are perpendicular if and only if their slopes multiply to $-1$.

Real-World Applications

Creating Cost Models from Data

Businesses often need to create pricing equations from known data points. If you know that producing 100 items costs $500 and producing 300 items costs $900, you can find the linear cost equation:

1. Calculate slope: $m = \frac{900 - 500}{300 - 100} = \frac{400}{200} = 2$ dollars per item
2. Use point-slope form: $y - 500 = 2(x - 100)$
3. Simplify: $y = 2x + 300$

The equation tells you: each item costs $2 to produce (variable cost), plus there is a $300 fixed cost regardless of production level.

Predicting Future Values

Linear equations let you extrapolate trends. If a company’s revenue was $1.2 million in 2020 and $1.8 million in 2023, and growth is approximately linear:

1. Slope: $\frac{1.8 - 1.2}{2023 - 2020} = \frac{0.6}{3} = 0.2$ million per year
2. Equation: $y - 1.2 = 0.2(x - 2020)$ or $y = 0.2x - 402.8$

To predict 2025 revenue: $y = 0.2(2025) - 402.8 = 405 - 402.8 = 2.2$ million dollars.

Converting Between Units

Many unit conversions are linear. The formula $F = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. If you only knew two data points (water freezes at 0C/32F and boils at 100C/212F), you could derive this equation:

1. Slope: $\frac{212 - 32}{100 - 0} = \frac{180}{100} = \frac{9}{5}$
2. Y-intercept: When $C = 0$, $F = 32$, so $b = 32$
3. Equation: $F = \frac{9}{5}C + 32$

Modeling Relationships in Science

In physics, Ohm’s Law states that voltage equals current times resistance: $V = IR$. If resistance is constant, this is a linear relationship between voltage and current with slope equal to the resistance. Scientists use experimental data points to determine the resistance of a material by finding the slope of the voltage-current graph.

Self-Test Problems

Summary

• When you have slope and y-intercept, use slope-intercept form directly: $y = mx + b$.
• When you have slope and any point, use point-slope form: $y - y_1 = m(x - x_1)$, then simplify.
• When you have two points, first calculate the slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$, then use point-slope form.
• When reading from a graph, identify two clear points for slope and find where the line crosses the y-axis.
• Parallel lines have the same slope. To write a parallel equation, keep the slope and use the new point.
• Perpendicular lines have slopes that are negative reciprocals (flip and change sign). Their slopes multiply to $-1$.
• In real-world problems, rates of change become slopes and starting values become y-intercepts.
• Point-slope form is your most versatile tool because it works whenever you know the slope and at least one point.