A Linear Equation in Two Variables Is of the Form

Math 161 Notes

Section 2.3 Lines

A linear equation in two variables is of the form

When graphed, the function will be a STRAIGHT LINE.

Look at the following diagram of a staircase,

To determine the proper height of a staircase, the “slope” of the staircase matters. We must use the same “run” for each step and the same “rise” for each step and this determines the “steepness” of the staircase.

Formula for Calculating the SLOPE OF A LINE given two points.

Example: Find the slope of the line that goes through the two points and interpret the slope in two different ways.

Interpretation:

Facts about the slope of a line:

1) When the slope of a line is POSITIVE, the line slants UPWARD from left to right.

2)When the slope of a line is NEGATIVE, the line slants DOWNWARD, from left to right.

3)When the slope of a line is 0, the line is a HORIZONTAL LINE.

4)When the slope of a line is undefined, the line is a VERTICAL LINE.

Drawing graphs of lines:

When given a point and a slope, we graph by plotting the point first then use rise over run to plot the second point, third point etc.

Example: Draw the graph of the line given certain criteria:

a) Point (3, -2) and slope of 1/3.

b)Point (2,3) and undefined slope. (Use the same graph as above.)

c)Point (1, - 4) and slope of 0. (Use the same graph as above.

What do we need to find on the graph to do this?

Finding equations of lines when given:

a)Two points on the graph:

Step 1: Find the slope of the line that goes through the two points.

Step 2: Use the point slope form of an equation to get the new equation.

POINT – SLOPE FORM AN EQUATION:

Example: Determine the equation of the line the is in the graph above.

SLOPE INTERCEPT FORM OF AN EQUATION:

The slope intercept form of an equation is of the form:

Example: Identify the slope and y – intercept of the given equations below;

a)b)

STANDARD FORM OF A LINEAR EQUATION:

How to graph a line when given an equation in standard form:

-Use the X and Y intercepts: What are these?

-X – intercept: the point where the graph crosses x – axis (x, 0)

-Y – intercept: the point where the graph crosses y – axis (0, y).

Example: Graph the equation using the X and Y intercepts only.

PARALLEL AND PERPENDICULAR LINES:

Parallel lines have SAME SLOPES

PERPENDICULAR LINES:

Perpendicular lines have OPPOSITE RECIPROCALSLOPES.

Example: Determine whether the given lines are parallel, perpendicular or neither: