**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.
- Determine lines parallel or perpendicular to given lines.

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 .

Slope describes the ** rate of change** of with respect to .

- 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?