Graphing and Analyzing Equations with Two Variables

Graphs, Functions, and Models

Solving Two Variable Equations

In College Algebra, many equations involve two variables, typically x and y. Understanding how to solve and graph these equations is foundational for analyzing mathematical relationships.

• Equations with One Variable: An equation like $x + 2 = 5$ has a single solution, $x = 3$, which can be represented as a single point on a one-dimensional number line.

• Equations with Two Variables: An equation like $x + y = 5$ has infinitely many solutions, each represented as an ordered pair $(x, y)$ that satisfies the equation. These solutions form a line or curve on a two-dimensional coordinate plane.

• Checking Solutions: To determine if a point $(x, y)$ satisfies an equation, substitute the values into the equation and check if the statement is true.

• Graph of an Equation: The graph is a visual representation of all $(x, y)$ pairs that make the equation true. Points on the graph satisfy the equation; points not on the graph do not.

Example: For the equation $x + y = 5$, the points (3,2), (4,1), (0,5), and (–1,6) all satisfy the equation and lie on its graph.

Graphing Two Variable Equations by Plotting Points

To graph an equation with two variables, calculate and plot ordered pairs that satisfy the equation.

1. Isolate y to the left side of the equation if possible.

2. Choose 3–5 values for x and calculate the corresponding y values.

3. Plot the resulting $(x, y)$ points on the coordinate plane.

4. Connect the points with a line or curve, as appropriate.

Example: To graph $-2x + y = 1$, create a table of values:

x	y	Ordered Pair (x, y)
-2	-3	(-2, -3)
-1	-1	(-1, -1)
0	1	(0, 1)
1	3	(1, 3)
2	5	(2, 5)

Plot these points and connect them to form the graph of the equation.

Practice: Graphing Nonlinear Equations

Some equations, such as quadratics or square roots, produce curves rather than straight lines.

• Quadratic Example: $y - x^2 + 3 = 0$ or $y = x^2 - 3$

• Square Root Example: $y = \sqrt{x} + 1$ (choose only non-negative x-values)

For each, create a table of values, plot the points, and connect them smoothly to reflect the curve.

Graphing by Plotting Points: Step-by-Step

1. Isolate y in the equation.

2. Choose 3–5 x-values and calculate corresponding y-values.

3. Plot the $(x, y)$ points from your table.

4. Connect the points with a line or curve.

Graphing Intercepts

Understanding Intercepts

Intercepts are points where a graph crosses the x-axis or y-axis. They are important for quickly sketching and analyzing graphs.

Type	Definition	How to Find
x-intercept	The x-value when the graph crosses the x-axis (y = 0)	Set y = 0 and solve for x
y-intercept	The y-value when the graph crosses the y-axis (x = 0)	Set x = 0 and solve for y

• At the x-intercept, the y-value is always zero.

• At the y-intercept, the x-value is always zero.

Example: For a given graph, the x-intercept might be at $x = -3.5$ and the y-intercept at $y = 4$.

Example: For another graph, the intercepts are at $(2, 0)$ and $(0, 1.3)$.

Summary Table: Intercepts

Intercept	Location	How to Find
x-intercept	Where the graph crosses the x-axis	Set y = 0, solve for x
y-intercept	Where the graph crosses the y-axis	Set x = 0, solve for y

Key Points

• Intercepts are useful for quickly sketching graphs and understanding the behavior of equations.

• Always check for intercepts when analyzing or graphing an equation.