1. 2.4 More about Linear Equations

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

3. 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.

4. 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 = Line 2: slope = 5-3 = /5 is the negative reciprocal of 5/2, so these lines are perpendicular.

5. 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.

6. 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

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

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

9. Standard Form Example: 4x - 8y = -24
Set up a T-chart
To find the X-intercept To find the y-intercept
Set y= 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

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

11. 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

12. 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

13. 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.

14. 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.

15. 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)

16. 3 Ways to Write the Equation of a Line

17. 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 ______.

18. 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)

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

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

21. Classwork
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