Linear Equations and Inequalities in Two Variables

Introduction to Linear Equations in Two Variables

Linear equations in two variables are equations that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. The graph of such an equation is always a straight line. Understanding how to graph these equations and identify their intercepts is fundamental in College Algebra.

Graphing Linear Equations Using Intercepts

One of the most efficient ways to graph a linear equation is by finding its intercepts. The intercepts are the points where the graph crosses the x-axis and y-axis.

x-intercept

• The x-intercept is the x-coordinate of the point where the graph crosses the x-axis.

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

• To find the x-intercept, set y = 0 in the equation and solve for x.

• Example: The graph crosses the x-axis at (2, 0). The x-intercept is 2.

y-intercept

• The y-intercept is the y-coordinate of the point where the graph crosses the y-axis.

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

• To find the y-intercept, set x = 0 in the equation and solve for y.

• Example: The graph crosses the y-axis at (0, 4). The y-intercept is 4.

Identifying Intercepts from a Graph

• To identify intercepts from a graph, locate the points where the line crosses the axes.

• Example 1a: The graph crosses the x-axis at (–3, 0) and the y-axis at (0, 5). Thus, the x-intercept is –3 and the y-intercept is 5.

• Example 1b: The graph crosses both axes at (0, 0). Thus, both the x-intercept and y-intercept are 0.

Finding Intercepts Algebraically

• To find the x-intercept of an equation such as , set y = 0 and solve for x:

• The x-intercept is 3, so the graph passes through (3, 0).

• To find the y-intercept, set x = 0 and solve for y:

• The y-intercept is –4, so the graph passes through (0, –4).

Steps for Graphing Ax + By = C Using Intercepts

1. Find the x-intercept (let y = 0 and solve for x).

2. Find the y-intercept (let x = 0 and solve for y).

3. Find a checkpoint (a third ordered-pair solution).

4. Draw a line through the three points to graph the equation.

Example: Graphing a Linear Equation Using Intercepts

• Graph .

• Find the x-intercept: Set y = 0.

• The x-intercept is 3, so the line passes through (3, 0).

• Find the y-intercept: Set x = 0.

• The y-intercept is 2, so the line passes through (0, 2).

• Find a checkpoint: Let x = 1.

• The checkpoint is (1, 4/3).

Graphing Equations of the Form Ax + By = 0

• When the constant term is zero, the line passes through the origin (0, 0).

• Find two additional points by choosing values for x or y and solving for the other variable.

• Example: For , let y = –1, then x = 3; let y = 1, then x = –3. The points (0, 0), (3, –1), and (–3, 1) are on the line.

Graphing Horizontal and Vertical Lines

• Horizontal lines have equations of the form y = k, where k is a constant. All points on the line have the same y-value.

• Vertical lines have equations of the form x = h, where h is a constant. All points on the line have the same x-value.

• Example: The line y = 3 is horizontal and passes through all points with y = 3. The line x = –2 is vertical and passes through all points with x = –2.