Linear Equation and Example Essay

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Section 1.2

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:…