Introduction

Linear equations are one of the most important ideas in algebra. They describe simple relationships between two variables—usually $x$ and $y$.
In this article, we explore what these relationships look like, how to read them, and how to interpret the patterns they create.

What Is a Linear Equation?

A linear equation is an equation where:

• Each variable appears only to the first power.
• Variables are not multiplied together.
• Variables are not inside square roots, exponents, or denominators.

Examples of linear equations:

• $y = 2x + 3$
• $3x - y = 5$
• $x = 7$

Non-examples:

• $y = x^2 + 1$ (has $x^2$)
• $xy = 4$ (variables multiplied)
• $y = \frac{1}{x}$ (variable in denominator)

A linear equation in two variables describes a straight line.

Slope and Intercept

A very common way to write a linear equation is: $$y = mx + b$$ Where:

• $m$ is the slope
• $b$ is the y-intercept

Slope ($m$)

The slope tells you:

• How steep the line is.
• Whether it rises or falls.
• How much $y$ changes when $x$ increases by 1.

Interpretations:

• If $m > 0$: the line rises.
• If $m < 0$: the line falls.
• If $m = 0$: the line is horizontal.

Y-intercept ($b$)

The y-intercept is the value of $y$ when $x = 0$.

Examples:

• In $y = 2x + 3$, the line crosses the y-axis at $3$.
• In $y = -x + 1$, the line crosses at $1$.

Graphing Linear Equations

To graph a linear equation:

1. Find the y-intercept $(0, b)$.
2. Use the slope $m$ to find another point.
   • If $m = \frac{3}{2}$, move:
     • Right 2 units
     • Up 3 units
3. Draw a straight line through the points.

Key idea:
Two points determine a unique line.

Standard Form

Another common way to write a linear equation is: $$Ax + By = C$$ Where:

• $A$, $B$, and $C$ are numbers.
• $A$ and $B$ are not both zero.

This form is useful for:

• Quickly finding intercepts.
• Describing vertical lines (which slope-intercept form cannot do).

Example:

• $3x + 2y = 12$

To find intercepts:

• Set $x = 0$ to find the y-intercept.
• Set $y = 0$ to find the x-intercept.

Interpreting Linear Relationships

Linear equations describe relationships where:

• A change in $x$ always produces a consistent change in $y$.
• The rate of change is constant (the slope).

Examples of real-world linear relationships:

• Cost = (price per item)$\times$(number of items) + (fixed fee)
• Distance traveled at constant speed
• Temperature conversion between Celsius and Fahrenheit

Exercises

1. Identify the slope and y-intercept of the equation $y = -3x + 5$

   Solution

   Slope: $-3$
   Y-intercept: $5$
2. Rewrite the equation $2x + y = 7$ in slope-intercept form.

   Solution

   Starting with $2x + y = 7$:
   Subtract $2x$: $$y = -2x + 7$$
3. Determine whether the equation $4y = 8x - 12$ is linear.

   Solution

   Yes, it is linear.
   Rewrite: $$4y = 8x - 12 \Rightarrow y = 2x - 3$$ No exponents, no products of variables.
4. Find the x-intercept of the equation $3x - 6 = 0$.

   Solution

   Solve $3x - 6 = 0$: $$3x = 6 \Rightarrow x = 2$$ The x-intercept is $(2, 0)$.
5. Graph the line $y = \frac{1}{2}x - 4$ by identifying two points.

   Solution

   

6. True or false: The equation $y = x^2 + 1$ is linear.

   Solution

   False.
   The equation contains $x^2$, so it is not linear.
7. Convert the equation $x = 9$ into standard form.

   Solution

   $x = 9$ can be written as: $$1x + 0y = 9$$
8. For the equation $y = 5$, describe the slope and the type of line.

   Solution

   $y = 5$ is a horizontal line.

   • Slope: $0$
   • It crosses the y-axis at $(0, 5)$.