Linear Equations

What Is a Linear Equation?

A linear equation is an algebraic equation that, when graphed on a coordinate plane, forms a perfectly straight line. This is their defining characteristic and what makes them so predictable and useful. These equations describe a relationship between variables where the rate of change is constant.

At its core, a linear equation involves one or two variables, and none of these variables have an exponent greater than 1. You won't find any terms like $x^2$, $\sqrt{x}$, or $\frac{1}{x}$ in a linear equation. For example, $y=2x+1$ is a linear equation. For every single step you take in the $x$ direction, you take a constant two steps in the $y$ direction. An equation like $y=x^2-3$ is not linear because the $x$ is squared, which creates a curved line (a parabola) when graphed.

Linear equations are the building blocks of algebra and appear in many real-world situations. They can have one variable, like $5x-10=0$, or two variables, like the more common $y=3x-5$. Understanding them is the first major step toward mastering algebra.

The Three Main Forms of Linear Equations

To work with linear equations effectively, we need to know how to write them. Mathematicians have developed three primary forms, each with its own strengths. Knowing how to recognize, use, and convert between these forms is a critical skill.

1. Slope-Intercept Form: $y=mx+b$

This is often called the most popular form because it's so intuitive for graphing. The variable $m$ represents the slope of the line (its steepness), and $b$ represents the y-intercept, which is the point where the line crosses the vertical y-axis.

2. Point-Slope Form: $y-y_1=m(x-x_1)$

This form is your best friend when you know the slope of a line and a single point that it passes through. In the formula, $m$ is the slope, and $(x_1,y_1)$ are the coordinates of the known point. You can easily rearrange this form into slope-intercept form with a little algebra.

3. Standard Form: $Ax+By=C$

Standard form places the $x$ and $y$ terms on the left side of the equation and a constant on the right. It's less visual than slope-intercept form but is useful for certain types of problems, like easily finding the x- and y-intercepts. To find the y-intercept, set $x=0$ and solve for $y$. To find the x-intercept, set $y=0$ and solve for $x$.

What Is Slope and How Is It Calculated?

The slope of a line is a number that measures its steepness and direction. It's often described as "rise over run." The "rise" is the vertical change between two points, and the "run" is the horizontal change between those same two points. A larger slope value means a steeper line.

To calculate the slope, you need any two points on the line. Let's call them $(x_1,y_1)$ and $(x_2,y_2)$. The formula for the slope, represented by the variable $m$, is:

There are four types of slope you'll encounter:

• Positive Slope: The line goes uphill from left to right.
• Negative Slope: The line goes downhill from left to right.
• Zero Slope: The line is perfectly flat (horizontal). The rise is $0$.
• Undefined Slope: The line is perfectly straight up and down (vertical). The run is $0$, which leads to division by zero in the formula.

How Do You Graph a Linear Equation?

Graphing a linear equation makes the abstract algebra visual. While you can always plot points by plugging in various $x$-values and solving for $y$, the fastest method is to use the slope-intercept form, $y=mx+b$.

Here is a step-by-step guide to graphing using this form:

1. Isolate y: Make sure your equation is in slope-intercept form. If it's in another form, use algebra to solve for $y$.
2. Plot the Intercept: Find the y-intercept, $b$, on the y-axis. This is your starting point. Remember, its coordinates are $(0,b)$.
3. Use the Slope: From your y-intercept, use the slope, $m$, to find a second point. Remember that $m=\frac{\text{rise}}{\text{run}}$. If the slope is positive, you go up (rise) and right (run). If it's negative, you go down (rise) and right (run).
4. Draw the Line: Use a ruler to draw a straight line that passes through your two points. Add arrows to both ends to show that the line continues infinitely.

How Do You Write the Equation of a Line?

Sometimes you'll be given information about a line and asked to write its equation. The form you start with depends on the information you're given.

Scenario 1: You are given the slope and the y-intercept.

This is the most straightforward case. Simply substitute the slope for $m$ and the y-intercept for $b$ directly into the slope-intercept form $y=mx+b$.

Scenario 2: You are given the slope and one point.

This is the perfect time to use point-slope form: $y-y_1=m(x-x_1)$. You substitute the given slope for $m$ and the coordinates of the point for $(x_1,y_1)$. Often, you will then be asked to convert the result to slope-intercept form.

Scenario 3: You are given two points.

If you have two points, you can first use the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ to find the slope. Once you have the slope, this scenario becomes identical to Scenario 2. Pick one of the original points and use point-slope form to find the equation.

What About Horizontal and Vertical Lines?

Horizontal and vertical lines are special cases of linear equations. They can seem tricky because one of the variables is missing from their simplified equation.

Horizontal Lines

A horizontal line has a slope of $0$. Every point on the line has the exact same y-coordinate. Because the slope $m=0$, the equation $y=mx+b$ simplifies to $y=(0)x+b$, which becomes just $y=b$. The equation of a horizontal line is always in the form $y=c$, where $c$ is the y-coordinate of every point on the line.

Vertical Lines

A vertical line has an undefined slope. Every point on this line has the exact same x-coordinate. Because the slope is undefined (due to division by zero in the slope formula), we cannot use slope-intercept or point-slope form. The equation of a vertical line is always in the form $x=c$, where $c$ is the x-coordinate of every point on the line.

Common Mistakes to Avoid

As you work with linear equations, some common pitfalls can trip you up. Being aware of them is the first step to avoiding them!

• Mixing up Slope Formula: A very common error is to put the change in $x$ in the numerator of the slope formula. Remember, it's always rise ($y$-values) over run ($x$-values).
• Sign Errors with Negatives: When subtracting negative numbers, as in $5-(-2)$, remember that it becomes addition ($5+2$). This is especially common in the slope and point-slope formulas.
• Confusing Horizontal and Vertical Lines: Students often mix up $y=c$ and $x=c$. A good way to remember is that $y=c$ has a $0$ slope and is flat, while $x=c$ is straight up and down with an undefined slope.
• Forgetting to Distribute: In point-slope form, $y-y_1=m(x-x_1)$, make sure you distribute the slope $m$ to both the $x$ and the $-x_1$ inside the parentheses.
• Incorrectly Identifying the Y-Intercept: The $b$ value is only the y-intercept when the equation is in the form $y=mx+b$. If the equation is $2y=4x+6$, you must first divide everything by $2$ to get $y=2x+3$ before you can say the y-intercept is $3$.

Quick Reference and Summary

This lesson covered the fundamentals of linear equations. Here are the most important takeaways to remember:

• A linear equation graphs a straight line and has variables with no exponent higher than 1.
• Slope measures the steepness of a line.
• The three main forms provide different ways to write and analyze the equation of a line.
• You can graph a line quickly using its slope and y-intercept.
• Horizontal lines have the form $y=c$ and a slope of 0.
• Vertical lines have the form $x=c$ and an undefined slope.

Key Formulas