3.1 Graph Linear Equations in Two Variables

The coordinate plane is a tool for visualizing mathematical relationships. It lets you plot points, graph equations, and see how variables relate to each other. Understanding how to work with coordinates and graph lines is a core skill in algebra that shows up constantly in later courses.

This section covers how to plot points, graph linear equations using several methods, and recognize special relationships between lines. Getting comfortable with these techniques now will pay off when you hit systems of equations, inequalities, and eventually calculus.

Graphing in the Coordinate Plane

Points in rectangular coordinates

The coordinate plane is a two-dimensional surface formed by two number lines crossing at right angles. The horizontal line is the x-axis, the vertical line is the y-axis, and they meet at the origin, which has coordinates (0, 0).

Every point on the plane is described by an ordered pair (x, y). The x-coordinate (sometimes called the abscissa) tells you how far to move left or right from the origin. The y-coordinate (sometimes called the ordinate) tells you how far to move up or down.

To plot a point like (3, −2), start at the origin, move 3 units right, then 2 units down.

The axes divide the plane into four quadrants, numbered counterclockwise:

• Quadrant I: x > 0, y > 0 (upper right)
• Quadrant II: x < 0, y > 0 (upper left)
• Quadrant III: x < 0, y < 0 (lower left)
• Quadrant IV: x > 0, y < 0 (lower right)

Points that sit directly on an axis don't belong to any quadrant. For example, (0, 5) is on the y-axis, not in Quadrant I or II.

Methods for graphing linear equations

A linear equation in two variables has the general form $ax + by = c$, where $a$, $b$, and $c$ are real numbers and $a$ and $b$ are not both zero. The graph of any linear equation is a straight line, which means you only need two points to draw it. There are three common methods for finding those points.

Point-plotting method:

1. Pick any value for $x$ and solve for $y$ (or vice versa) to get an ordered pair.
2. Repeat with a different value to get a second point. Picking a third point as a check is a good habit.
3. Plot the points and draw a straight line through them.

For example, given $2x + y = 6$: if $x = 0$, then $y = 6$; if $x = 1$, then $y = 4$. Plot (0, 6) and (1, 4), then connect them.

Intercept method:

1. Find the x-intercept by setting $y = 0$ and solving for $x$.
2. Find the y-intercept by setting $x = 0$ and solving for $y$.
3. Plot both intercepts and draw the line through them.

This is often the fastest method when the equation is in standard form. Using the same example, $2x + y = 6$: the x-intercept is (3, 0) and the y-intercept is (0, 6).

Slope-intercept method:

1. Rewrite the equation in slope-intercept form: $y = mx + b$.
2. Plot the y-intercept at (0, $b$).
3. From that point, use the slope $m$ (rise over run) to locate a second point.
4. Draw the line through both points.

For $2x + y = 6$, rewrite as $y = -2x + 6$. The y-intercept is (0, 6) and the slope is $-2$, which means you go down 2 units and right 1 unit to reach the next point, (1, 4).

Vertical vs. horizontal lines

These are two special cases that don't fit neatly into slope-intercept form.

• Vertical lines have the equation $x = a$, where $a$ is a constant. Every point on the line has the same x-coordinate. A vertical line is parallel to the y-axis, and its slope is undefined (because the run is zero, and you can't divide by zero).
• Horizontal lines have the equation $y = b$, where $b$ is a constant. Every point on the line has the same y-coordinate. A horizontal line is parallel to the x-axis, and its slope is zero (the line doesn't rise at all).

A common mistake is mixing these up. Remember: $x = 3$ is a vertical line passing through (3, 0), while $y = 3$ is a horizontal line passing through (0, 3).

Intercepts and Graphing Techniques

Intercepts of linear equations

• The x-intercept is the point where the graph crosses the x-axis. At this point, $y = 0$. To find it, substitute $y = 0$ into the equation and solve for $x$.
• The y-intercept is the point where the graph crosses the y-axis. At this point, $x = 0$. To find it, substitute $x = 0$ into the equation and solve for $y$.

Write intercepts as ordered pairs. For $3x - 4y = 12$: setting $y = 0$ gives $x = 4$, so the x-intercept is (4, 0). Setting $x = 0$ gives $y = -3$, so the y-intercept is (0, −3).

One thing to watch: if the line passes through the origin, both intercepts are (0, 0). In that case, the intercept method only gives you one point, so you'll need to find a second point by plugging in a value for $x$ or $y$.

Efficient line graphing techniques

You should be comfortable recognizing and using all three forms of a linear equation:

Slope-intercept form $y = mx + b$

• $m$ is the slope and $b$ is the y-intercept.
• This is the most convenient form for graphing because you can read the slope and y-intercept directly.

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

• $m$ is the slope and $(x_1, y_1)$ is a specific point on the line.
• This form is most useful when you know the slope and one point (but that point isn't the y-intercept). Plot the known point, then use the slope to find a second point.

Standard form $ax + by = c$

• $a$, $b$, and $c$ are constants.
• The intercept method works well here: find both intercepts, plot them, and draw the line.

Whichever form you start with, you can always convert to another. The most common conversion is rearranging standard form into slope-intercept form by isolating $y$.

Slope and Special Line Relationships

Slope of a line

Slope measures the steepness of a line and the rate at which $y$ changes relative to $x$. It's calculated as "rise over run":

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

Pick any two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$, and plug them in. The order doesn't matter as long as you're consistent (don't subtract the first y from the second and then the second x from the first).

For example, the slope between (1, 2) and (4, 8) is $m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$.

What the sign of the slope tells you:

• Positive slope: the line rises from left to right.
• Negative slope: the line falls from left to right.
• Zero slope: the line is horizontal.
• Undefined slope: the line is vertical.

Parallel and perpendicular lines

Parallel lines never intersect and have the same slope. If one line has slope $m = 3$, any line parallel to it also has slope 3. Their y-intercepts will be different (otherwise they'd be the same line).

Perpendicular lines intersect at a 90° angle. Their slopes are negative reciprocals of each other. That means if one line has slope $m$, the perpendicular line has slope $-\frac{1}{m}$.

For example, if a line has slope $\frac{2}{3}$, a perpendicular line has slope $-\frac{3}{2}$. You flip the fraction and change the sign.

A quick check: multiply the two slopes together. If the product is $-1$, the lines are perpendicular. ($\frac{2}{3} \times -\frac{3}{2} = -1$ ✓)

Note that vertical and horizontal lines are perpendicular to each other, but this pair is a special case since vertical lines have undefined slope, so the negative reciprocal rule doesn't apply directly.