Copyright © 2011 Pearson Education, Inc.

Linear Equations in Two Variables

Section 1.4

Equations, Inequalities, and Modeling

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1.4

The steepness or slope of a line in the xy-coordinate system is the ratio of the rise (the change in y-coordinates) to the run (the change in x-coordinates) between two points on the line.

Note that that if (x1, y1) and (x2, y2) are two points for which x1 = x2, then the line through them is a vertical line. Since this case is not included in the definition of slope, a vertical line does not have a slope. We also say that the slope of a vertical line is undefined.

If we choose two points on a horizontal line, then y1 = y2 and y2 – y1 = 0. For any horizontal line the rise between two points is 0 and the slope is 0.

run

rise

scoordinate- in change

scoordinate- in changeslope

x

y

x

y

Slope of a Line

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Definition: SlopeThe slope of the line through (x1, y1) and (x2, y2) with x1 ≠ x2 is

.12

12

xx

yy

Slope

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1.4

Suppose that a line through (x1, y1) has slope m. Every other

point (x, y) on the line must satisfy the equation

because any two points can be used to find the slope. Multiply both sides by x – x1 to get y – y1 = m(x – x1), which is the

point-slope form of the equation of the line.

mxxyy

1

1

Theorem: Point-Slope FormThe equation of the line (in point-slope form) through (x1, y1)with slope m is

y – y1 = m(x – x1).

Point-Slope Form

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The line y = mx + b goes through (0, b) and (1, m + b).

Between these two points the rise is m and the run is 1. So the

slope is m. Since (0, b) is the y-intercept and m is the slope,

y = mx + b is called the slope-intercept form. Any equation in

standard form Ax + By = C can be rewritten in slope-intercept

form by solving the equation for y, provided that B ≠ 0.

Theorem: Slope-Intercept FormThe equation of the line (in slope-intercept form) with slope mand y-intercept (0, b) is

y = mx + b.Every nonvertical line has an equation in slope-intercept form.

Slope-Intercept Form

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Strategy: Finding the Equation of a Line Standard from Ax + By = C Slope-intercept form y = mx + b

Point-slope form y – y1 = m(x – x1)

1. Since vertical lines have no slope, they can’t be written in slope-intercept or point-slope form.

2. All lines can be described with an equation in standard form.

3. For any constant k, y = k is a horizontal line and x = k is a vertical line.

4. If you know two points on a line, then find the slope.

5. If you know the slope and point on the line, use the point-slope form. If the point is the y-intercept, then use the slope-intercept form.

6. Final answers are usually written in slope-intercept or standard form. Standard form is often simplified by using only integers for the coefficients.

The Three Forms forthe Equation of a Line

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Theorem: Parallel LinesTwo non-vertical lines in the coordinate plane are parallel ifand only if their slopes are equal.

Theorem: Perpendicular LinesTwo lines with slopes m1 and m2 are perpendicular if and

only if m1m2 = –1.

Parallel and Perpendicular Lines