Graphing and Finding Intercepts of Linear Equations, Slides of Algebra

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§2.4a Lines

by Intercepts

Review §

 Any QUESTIONS About

• § 2.3 → Algebra of Funtions

 Any QUESTIONS About HomeWork

• § 2.2 → HW-

2.3 MTH 55

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

4 y + 3 x = 18 4•5 + 3•1 | 18 20 + 3 | 18 23 = 18  False

To Graph a Linear Equation

1. 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.
2. Repeat step (1) to find a second ordered pair. A third ordered pair can be used as a check.
3. Plot the ordered pairs and draw a straight line passing through the points. The line represents ALL solutions of the equation

Example  Graph y = − 4 x + 1

• Results are often listed in a table. x y ( x, y ) 2 –7 (2, –7) 0 1 (0, 1) –2 9 (–2, 9)
• Choose x
• Compute y.
• Form the pair ( x , y ).
• Plot the points.

Example  Graph y = − 4 x + 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 = − 4 x + 1

Example Graph 4 y = 3 x

• Solution : Begin by solving for y. 4 y = 3 x

y 3 x 4

1 4 4

1 ⋅ = ⋅

y x 0. 75 x 4

3 = =

 Or y is 75% of x

 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

Example  Graph 4 y = 3 x

• 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 (0, 0) 4 3 (4, 3) − 4 − 3 (−4 , −3) 3 4

y = x

FedEx Soln: c = 2.8 w + 21.

• Select values for w and then calculate c.
• c = 2.8 w + 21.
  • If w = 2, then c = 2.8(2) + 21.05 = 26.
  • If w = 4, then c = 2.8(4) + 21.05 = 32.
  • If w = 8, then c = 2.8(8) + 21.05 = 43.
• Tabulating the Results:

w c 2 26. 4 32. 8 43.

FedEx Soln: Graph Eqn

• Plot the points.

Weight (in pounds)

Mail cost (in dollars)

 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

 The cost of shipping an^ 10 ½ pounds 10½ pound package is about $51.

≈ $

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)

Graph Ax + By = C Using Intercepts

1. Find the x -Intercept  Let y = 0, then solve for x
2. Find the y -Intercept  Let x = 0, then solve for y
3. Construct a CheckPoint using any convenient value for x or y
4. Graph the Equation by drawing a line thru the 3-points (i.e., connect the dots)

Example  Find Intercepts

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

Example  Find Intercepts cont.

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