Lesson 26: Writing the Equation of a Line

New Concept

If you know the slope $m$ and an ordered pair $(x_1, y_1)$ of any point on the line, you can use the point-slope form to write the equation of the line.

Why it matters

Algebra is the language used to describe relationships, and linear equations are its most fundamental sentences. Mastering how to write these equations from any given information is the key to unlocking mathematical modeling.

What’s next

Next, you’ll apply these forms to solve problems, from converting temperatures to modeling real-world business and growth data.

The standard form of a linear equation is written as $Ax + By = C$. For this equation, $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero.

Given $y = \frac{2}{3}x - 5$. Step 1: Move the x-term to get $-\frac{2}{3}x + y = -5$. Step 2: Multiply by $-3$ to clear the fraction and make A positive: $2x - 3y = 15$. The equation $y = -0.5x + 10$ becomes $0.5x + y = 10$, and then $x + 2y = 20$.

Think of this as the 'tidy' form where variables hang out on one side and the constant chills on the other. It's great for seeing relationships cleanly, even if it hides the slope. It's also the final boss of formatting, with no fractions allowed!

The slope-intercept form of a linear equation is written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

A line with slope 3 and y-intercept at $(0, 7)$ is simply $y = 3x + 7$. Given $m = -5$ and a y-intercept of 12, the equation is immediately written as $y = -5x + 12$. A line that crosses the y-axis at -1 with a slope of 1 is $y = x - 1$.

This is the 'ready-to-graph' form! It’s the ultimate cheat code for graphing because it tells you exactly where to start on the y-axis (that's your 'b') and which direction to travel (that's your slope 'm'). Just plot the point and follow the slope!

If you know the slope $m$ and an ordered pair $(x_1, y_1)$ of any point on the line, then you can use the point-slope form to write the equation of the line: $y - y_1 = m(x - x_1)$.

A line has slope $-2$ and passes through $(10, 12)$. Setup: $y - 12 = -2(x - 10)$. Solving gives $y = -2x + 20 + 12$, which simplifies to $y = -2x + 32$. For a line with slope 3 through $(-4, 1)$: $y - 1 = 3(x - (-4))$, which simplifies to $y = 3x + 13$.

Got a point and a slope? This formula is your best friend. It’s the starting block for building the line's full equation when you don't know the y-intercept right away. Just plug in what you know, solve for y, and you’re golden!