1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§2.4a Lines by Intercepts
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 2.3 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§ 2.3 → Algebra of Funtions
Any QUESTIONS About HomeWork
§ 2.2 → HW-05

3. Eqn of a Line  Ax + By = C
Determine whether each of the following pairs is a solution of eqn 4y + 3x = 18:
a) (2, 3); b) (1, 5).
Soln-a) We substitute 2 for x and 3 for y
4y + 3x = 18
4•3 + 3•2 | 18
| 18
18 = True
Since 18 = 18 is true, the pair (2, 3) is a solution

4. Example  Eqn of a Line Soln-b) We substitute 1 for x and 5 for y
Since 23 = 18 is false, the pair (1, 5) is not a solution
4y + 3x = 18
4•5 + 3•1 | 18
| 18
23 = 18  False

5. To Graph a Linear Equation
Select a value for one coordinate and calculate the corresponding value of the other coordinate. Form an ordered pair. This pair is one solution of the equation.
Repeat step (1) to find a second ordered pair. A third ordered pair can be used as a check.
Plot the ordered pairs and draw a straight line passing through the points. The line represents ALL solutions of the equation

6. Example  Graph y = −4x + 1
Solution: Select convenient values for x and compute y, and form an ordered pair.
If x = 2, then y = −4(2)+ 1 = −7 so (2,−7) is a solution
If x = 0, then y = −4(0) + 1 = 1 so (0, 1) is a solution
If x = –2, then y = −4(−2) + 1 = 9 so (−2, 9) is a solution.

7. Example  Graph y = −4x + 1 Results are often listed in a table. x y2 –7
(2, –7) 1
(0, 1) –2 9
(–2, 9)
Choose x
Compute y.
Form the pair (x, y).
Plot the points.

8. Example  Graph y = −4x + 1
Note that all three points line up. If they didn’t we would know that we had made a mistake
Finally, use a ruler or other straightedge to draw a line
Every point on the line represents a solution of: y = −4x + 1

9. Example  Graph x + 2y = 6
Solution: Select some convenient x-values and compute y-values.
If x = 6, then 6 + 2y = 6, so y = 0
If x = 0, then 0 + 2y = 6, so y = 3
If x = 2, then 2 + 2y = 6, so y = 2
In Table Form, Then Plotting x y
(x, y) 6
(6, 0) 3
(0, 3) 2
(2, 2)

10. Example Graph 4y = 3x Solution: Begin by solving for y.
To graph the last Equation we can select values of x that are multiples of 4
This will allow us to avoid fractions when computing the corresponding y-values
Or y is 75% of x

11. Example  Graph 4y = 3x
Solution: Select some convenient x-values and compute y-values.
If x = 0, then y = ¾ (0) = 0
If x = 4, then y = ¾ (4) = 3
If x = −4, then y = ¾ (−4) = −3
In Table Form, Then Plotting x y
(x, y)
(0, 0) 4 3
(4, 3) −4 −3
(4 , 3)

12. Example  Application
The cost c, in dollars, of shipping a FedEx Priority Overnight package weighing 1 lb or more a distance of 1001 to 1400 mi is given by c = 2.8w
where w is the package’s weight in lbs
Graph the equation and then use the graph to estimate the cost of shipping a 10½ pound package

13. FedEx Soln: c = 2.8w + 21.05 Select values for w and then calculate c.
If w = 2, then c = 2.8(2) = 26.65
If w = 4, then c = 2.8(4) = 32.25
If w = 8, then c = 2.8(8) = 43.45
Tabulating the Results: w c 2
26.65 4
32.25 8
43.45

14. FedEx Soln: Graph Eqn Plot the points.
To estimate costs for a 10½ pound package, we locate the point on the line that is above 10½ lbs and then find the value on the c-axis that corresponds to that point
$51
Mail cost (in dollars)
The cost of shipping an 10½ pound package is about $51.00
10 ½ pounds
Weight (in pounds)

15. Finding Intercepts of Lines
An “Intercept” is the point at which a line or curve, crosses either the X or Y Axes
A line with eqn Ax + By = C (A & B ≠ 0) will cross BOTH the x-axis and y-axis
The x-CoOrd of the point where the line intersects the x-axis is called the x-intercept
The y-CoOrd of the point where the line intersects the y-axis is called the y-intercept

16. Example  Axes Intercepts
For the graph shown
a) find the coordinates of any x-intercepts
b) find the coordinates of any y-intercepts
Solution
a) The x-intercepts are (−2, 0) and (2, 0)
b) The y-intercept is (0,−4)

17. Graph Ax + By = C Using Intercepts
Find the x-Intercept  Let y = 0, then solve for x
Find the y-Intercept  Let x = 0, then solve for y
Construct a CheckPoint using any convenient value for x or y
Graph the Equation by drawing a line thru the 3-points (i.e., connect the dots)

18. To FIND the Intercepts
To find the y-intercept(s) of an equation’s graph, replace x with 0 and solve for y.
To find the x-intercept(s) of an equation’s graph, replace y with 0 and solve for x.

19. Example  Find Intercepts
Find the y-intercept and the x-intercept of the graph of 5x + 2y = 10
SOLUTION: To find the y-intercept, we let x = 0 and solve for y
5 • 0 + 2y = 10
2y = 10
y = 5
Thus The y-intercept is (0, 5)

20. Example  Find Intercepts cont.
Find the y-intercept and the x-intercept of the graph of 5x + 2y = 10
SOLUTION: To find the x-intercept, we let y = 0 and solve for x
5x + 2• 0 = 10
5x = 10
x = 2
Thus The x-intercept is (2, 0)

21. Example  Graph w/ Intercepts
Graph 5x + 2y = 10 using intercepts
SOLUTION:
We found the intercepts in the previous example. Before drawing the line, we plot a third point as a check. If we let x = 4, then
5 • 4 + 2y = 10
y = 10
2y = −10
y = − 5
We plot Intercepts (0, 5) & (2, 0), and also (4 ,−5)
5x + 2y = 10
y-intercept (0, 5)
x-intercept (2, 0)
Chk-Pt (4,-5)

22. Example  Graph w/ Intercepts
Graph 3x − 4y = 8 using intercepts
SOLUTION: To find the y-intercept, we let x = 0. This amounts to ignoring the x-term and then solving. −4y = 8
y = −2
Thus The y-intercept is (0, −2)

23. Example  Graph w/ Intercepts
Graph 3x – 4y = 8 using intercepts
SOLUTION: To find the x-intercept, we let y = 0. This amounts to ignoring the y-term and then solving x = x = 8/3
Thus The x-intercept is (8/3, 0)

24. Example  Graph w/ Intercepts
Construct Graph for 3x – 4y = 8
Find a third point. If we let x = 4, then
3•4 – 4y = 8
12 – 4y = 8
–4y = –4
y = 1
We plot (0, −2), (8/3, 0), and (4, 1) and Connect the Dots
Chk-Pt Charlie
x-intercept
y-intercept
3x  4y = 8

25. Example  Graph y = 2
SOLUTION: We regard the equation y = 2 as the equivalent eqn: 0•x + y = 2.
No matter what number we choose for x, we find that y must equal 2.
y=2
Choose any number for x. x y
(x, y) 2
(0, 2)
Ask: “ what does this graph look like?” 4 2
(4, 2) −4 2
(−4 , 2)
y must be 2.

26. Example  Graph y = 2
Next plot the ordered pairs (0, 2), (4, 2) & (−4, 2) and connect the points to obtain a horizontal line.
Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis with y-intercept (0, 2)
y = 2
(0, 2)
(4, 2)
(4, 2)

27. Example  Graph x = −2
SOLUTION: We regard the equation x = −2 as x + 0•y = −2. We build a table with all −2’s in the x-column.
x = −2
x must be 2. x y
(x, y) −2 4
(−2, 4) 1
(−2, 1) −4
(−2, −4)
Ask: “what does THIS graph look like?”
Any number can be used for y.

28. Example  Graph x = −2
When we plot the ordered pairs (−2,4), (−2,1) & (−2, −4) and connect them, we obtain a vertical line
Any ordered pair of the form (−2,y) is a solution. The line is parallel to the y-axis with x-intercept (−2,0)
x = 2
(2, 4)
(2, 1)
(2, 4)

29. Linear Eqns of ONE Variable
The Graph of y = b is a Horizontal Line, with y-intercept (0,b)
The Graph of x = a is a Vertical Line, with x-intercept (a,0)

30. Example  Horiz or Vert Line
Write an equation for the graph
SOLUTION: Note that every point on the horizontal line passing through (0,−3) has −3 as the y-coordinate.
Thus The equation of the line is y = −3

31. Example  Horiz or Vert Line
Write an equation for the graph
SOLUTION: Note that every point on the vertical line passing through (4, 0) has 4 as the x-coordinate.
Thus The equation of the line is x = 4

32. SLOPE Defined
The SLOPE, m, of the line containing points (x1, y1) and (x2, y2) is given by

33. Example  Slope City
Graph the line containing the points (−4, 5) and (4, −1) & find the slope, m
SOLUTION
Change in y = −6
Change in x = 8
Chg in y as x changes from LEFT to RIGHT
Thus Slope m = −3/4

34. Example  ZERO Slope Find the slope of the line y = 3
SOLUTION: Find Two Pts on the Line
(3, 3)
(2, 3)
Then the Slope, m
A Horizontal Line has ZERO Slope

35. Example  UNdefined Slope
Find the slope of the line x = 2
(2, 4)
SOLUTION: Find Two Pts on the Line
Then the Slope, m
(2, 2)
A Vertical Line has an UNDEFINED Slope

36. Applications of Slope = Grade
Some applications use slope to measure the steepness.
For example, numbers like 2%, 3%, and 6% are often used to represent the grade of a road, a measure of a road’s steepness.
That is, a 3% grade means that for every horizontal distance of 100 ft, the road rises or falls 3 ft.

37. Grade Example Find the slope (or grade) of the treadmill
SOLUTION: Noting the Rise & Run
0.42 ft
5.5 ft
In %-Grade for Treadmill

38. Slope Symmetry
We can Call EITHER Point No.1 or No.2 and Get the Same Slope
Example, LET
(x1,y1) = (−4,5)
(−4,5) Pt1
(4,−1)
Moving L→R

39. Slope Symmetry cont Now LET Moving R→L Thus (−4,5) (x1,y1) = (4,−1)
(4,−1) Pt1
Moving R→L
Thus

40. Slopes Summarized
POSITIVE Slope
NEGATIVE Slope

41. Slopes Summarized ZERO Slope UNDEFINED Slope slope = 0
        
slope = 0
slope = undefined
Note that when a line is horizontal the slope is 0
Note that when the line is vertical the slope is undefined

42. WhiteBoard Work Problems From §2.4 Exercise Set More Lines
26 (PPT), 12, 24, 52, 56
More Lines

43. P2.4-26  Find Slope for Lines
Recall

44. Some Slope Calcs All Done for Today

45. 20x20 Grid

46. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –