2.4 More about Linear Equations

Parallel Lines – have the same slope These lines are parallel -- The rise/run ratio of both lines is the same

Perpendicular Lines – the lines have slopes that are negative reciprocals of each other Then the slope of any perpendicular line must be 3/2 If this line has a slope of –2/3 ….. On any two perpendicular lines, the product of the slope is always -1.

Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (3,2) and (5,7) Line 2: through (0,3) and (-5,5) Find the slope of each line, then compare…. Line 1: slope = 7-2 = 5 5-3 2 Line 2: slope = 5-3 = 2 -5-0 -5 -2/5 is the negative reciprocal of 5/2, so these lines are perpendicular.

Standard Form of a Linear Equation Ax + By = C A and B are not both zero (usually integers). When the equation is in this form, we can graph it by finding the x and y intercepts.

X and Y intercepts are the points where a graph crosses the x-axis and the y-axis This is the y-intercept. At this point, x = 0 This is the x-intercept. At this point, y =0

Standard Form Example: 3x + 2y = 6 Set up a T-chart To find the X-intercept To find the y-intercept Set y= 0 Set x = 0______ 3x + 2(0) = 6 3x + 0 = 6 X = 2 3(0) + 2y = 6 0 + 2y = 6 y = 3

The x-intercept is (2,0) The y-intercept is (0,3)

Standard Form Example: 4x - 8y = -24 Set up a T-chart To find the X-intercept To find the y-intercept Set y= 0 Set x = 0______ 4x - 8(0) = -24 4x - 0 = -24 4x = -24 X = -6 4(0) - 8y = -24 0 - 8y = -24 -8y = -24 y = 3

The x-intercept is (-6,0) The y-intercept is (0,3)

Question: Does a linear equation always have a y-intercept? No. This line is vertical; it never crosses the y-axis. The equation for this line would be x = 5. Also, remember that a vertical line has NO SLOPE

Does a linear equation always have an x-intercept? NO. This line is horizontal; it never crosses the x-axis. The equation for this line would be y = 2. Also, remember that a horizontal line has a SLOPE = 0

Previously we learned two forms for equations of lines: Slope-intercept form……y = mx+b This gives us the slope (m) and the y-intercept (b) Standard Form: Ax + By = C In this form, we can find the x and y intercepts to graph the line.

Also gives us a point on the line: (x1, y1) Point-Slope Form y – y1 = m (x – x1) Gives us the slope: m Also gives us a point on the line: (x1, y1) You could have several equations for the same line, depending on the point that is used.

Write the equation of a line that passes through (2,4) with a slope of ½. Since we know the slope and a point on the line, begin by using point-slope form: y – y1 = m (x – x1) y – 4 = ½ (x – 2)

3 Ways to Write the Equation of a Line

Write the equation of a line that passes through (3,-1) and is perpendicular to the line y = 3x + 2. Recall that perpendicular lines have negative reciprocal slopes. The slope of the perpendicular line is ____. So, the slope of this line must be ______.

Passes through (3,-1); slope is -1/3 y – y1 = m (x – x1) y – (-1) = -1/3 (x-3) y + 1 = -1/3 (x-3)

Write the equation of a line that passes through (4,2) and (6,5) First, find the slope: m = y2 - y1 = 5-2 = 3 x2 - x1 6-4 2 Now, we can use the slope in our formula y – y1 = m (x – x1)

Passes through (4,2) and (6,5); slope is 3/2

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