A linear equation is of the form: ax + by = c (a, b, c are numbers)

Graphing a linear equation

* Find 2 or more (x, y) points that satisfy the equation

* Plot points

* Connect with straight line

Example: (#10) y = 5/3x – 3 (when x=-3, y = (5/3)(-3) – 3 = -5 – 3 = -8)

x | y |

-3 | -8 |

0 | -3 |

3 | 2 |

6 | 7 |

y-intercept: where the line crosses the y-axis (x=0)

x-intercept: where the line crosses the x-axis (y=0)

Graphing a line using intercepts

* Let x=0, solve for y y-intercept (0, )

* Let y=0, solve for x x-intercept ( , 0)

* Plot 2 intercepts and connect with line

* Plot a third point to check

Example: (#22) -3x -2y = 6

Let x=0: -3(0) – 2y = 6

0 – 2y = 6, divide both sides by (-2)

y = -3

y-intercept is (0, -3)

Let y=0: -3x -2(0) = 6

-3x = 6, divide both sides by (-3)

x = -2

x-intercept is (-2, 0)

Plot (0, -3) and (-2, 0) and connect

Slope of a Line

Let (x1, y1) and (x2, y2) be two points on a line, where x1 ≠ x2.

Slope m = y2 – y1 = change in y = ∆y (Greek letter ∆ means “change in”)

x2 – x1 change in x ∆x

Example: (#36)

(a) Find the slope of the line that goes through (-3, -1) and (0, 7).

∆y / ∆x = (-1 – 7) / (-3 – 0) = (-8) / (-3) = 8/3

(b) Find 2 additional points on the line (3, ) and (-6, ).

Slope of 8/3 means that as x increases 3, y increases 8.

So adding (3, 8) or any multiple of (3, 8) will also be on the line.

(0, 7) + ( 3, 8) = (3, 15), a point on the line

(-3,-1) + (-3,-8) = (-6, -9), a point on the line

Things to watch out for when computing slope:

* Mixing up the order: y2 – y1 is wrong

x1 – x2

* Change in y must be in top of fraction

* Be careful with negatives: -3 – (-1) = -3 + 1 = -2

Exercise #44 (reading a graph, finding slope)

Horizontal and Vertical Lines

Horizontal line: y = k slope = 0

Vertical line: x = h slope is undefined

Parallel and Perpendicular Lines

Let L1 and L2 be distinct, non-vertical lines.

L1 has slope m1 and L2 had slope m2.

“if and only if” may be abbreviated “iff” or

Parallel:

L1 and L2 are parallel m1 = m2 (slopes are equal)

Perpendicular:

L1 and L2 are perpendicular m1 • m2 = -1 (“sign and fraction flipped”)

Example: (#60) Use the slope formula to determine of the lines L1 and L2 are parallel,

perpendicular, or neither. L1: points (6, 2) and (8, -2)

L2: points (5, 1) and (3, 0)

Slope for L1: (2 – (-2)) / (6 – 8) = 4 / (-2) = -2

Slope for L2:…