**Section Objectives**

- Determine solutions of two-variable linear equations.
- Graph a line by finding two points on the line.
- Find the - and -intercepts of a line.
- Compute the slope of a line and interpret it as a rate of change.
- Identify equations of horizontal or vertical lines and graph them.

Any linear equation in two variables can be written in the form

where and are not both zero. The form is called ** standard form**.

A solution of a two-variable linear equation is a point that satisfies the equation. Every two-variable linear equation has infinitely many solutions. If all such solutions are plotted in a rectangular coordinate system, the resulting graph is a line.

- Show that both and are solutions of .

- Show that and are solutions of . Can you find two additional solutions?

- Find two solutions of .

Remember from geometry that a line is uniquely determined by two distinct points. Therefore, one approach (among others that we will learn) to graphing the line described by a two-variable linear equation is to:

- determine two solutions (i.e, two points on the line),
- plot them, and
- sketch a line through them.

Some of us may also like to find one more solution to check our work.

- Graph the line described by .

- Graph the line described by .

- Graph the line described by . What word would you use to describe the line?

- Graph the line described by . What word would you use to describe the line?

Points of the form and are called - and -intercepts, respectively. These are points where a graph intersects the -axis or the -axis.

In problem situations, Intercepts often have some kind of physical significance. They are usually worth finding! And they can be useful for graphing.

- Graph the line whose -intercept is and whose -intercept is .

- Determine the - and -intercepts. Then use them to graph the line.

The slant of a line is measured by a number called ** slope**. Before we define slope, here are three important ideas to keep in mind:

- Slope is only defined for non-vertical lines.
- The slope of a line is the same at any point on the line.
- The slope is a measure of the rate of change of with respect to .

Suppose and are any two distinct points on a non-vertical line. The ** slope** of the line is given by .

- Determine the slope of the line that passes through the two points and .

- Determine two points on the line described by the equation . Then use your points to find the slope of the line.

Two non-vertical lines are ** parallel** if and only if they have the same slope.

Two non-vertical lines are ** perpendicular** if and only if their slopes are opposite reciprocals (that is, the product of their slopes is ).

What if one or both of the lines are vertical?

- The line passes through the points and . Find the slope of a line parallel to . Find the slope of a line perpendicular to .

- The line passes through the points and . Find an equation of a line parallel to . Find an equation of a line perpendicular to .

- Write equations for the horizontal and vertical lines through .

- Sal fixes vintage arcade games. He charges a flat fee of $140 to make a house call, but then he charges a constant hourly rate on top of that. He recently made a house call to fix a Centipede game and ended up billing the a client $230 after 2 hours of work. Sketch the graph the shows how much Sal makes versus time (in hours). What does the slope of the graph represent?