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Topic 4.2.2Topic 4.2.2

Graphing Ax + By = CGraphing Ax + By = C

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Graphing Ax + By = CGraphing Ax + By = C

California Standards:6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

What it means for you:You’ll learn how to graph a straight line by joining two points.

Key Words:• linear equation

Topic4.2.2

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y = 2x

(2, 4)

(–2, –4)

Graphing Ax + By = CGraphing Ax + By = C

Every point on a line is a solution to the equation of the line.

If you know any two solutions (any two coordinate pairs), then you can join the points with a straight line.

Topic4.2.2

If you know any two solutions (any two coordinate pairs)…

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Graphing Ax + By = CGraphing Ax + By = C

Graphing the Line Ax + By = C Using Two Points

The graph of the equation Ax + By = C consists of all points (x, y) whose coordinates satisfy Ax + By = C.

To graph the line, you just need to plot two points on it and join them together with a straight line.

Topic4.2.2

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Graphing Ax + By = CGraphing Ax + By = C

• Rearrange the equation so it is in the form y = Px + Q.

• Choose two values of x and substitute them into your equation to find the corresponding values of y.

Topic4.2.2

• Plot the two points and draw a straight line through them.

• Plot a third point to check that the line is correct — the point should lie on the line.

Here’s how you go about it:

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Graphing Ax + By = CGraphing Ax + By = C

Example 1

Plot and label the graph of the equation x – y = –3.

Solution follows…

Topic4.2.2

Rearrange the equation to get y = x + 3.

Solution

Choose two values of x, then draw a table to help you find the y-values:

(4, 7)y = x + 3, so y = 4 + 3 = 74

(–2, 1)y = x + 3, so y = –2 + 3 = 1–2

(x, y)yx

Solution continues…

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Plot and label the graph of the equation x – y = –3.

Graphing Ax + By = CGraphing Ax + By = C

Example 1

When you plot the graph, the line should be straight.

Solution (continued)

Topic4.2.2

When x = 1

Check your solution:

So (x, y) = (1, 4)

y = x + 3 = 1 + 3 = 4

(1, 4) lies on the line — which means the line is correct.

(1, 4)(4, 7)

(–2, 1)

(x, y)

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Graphing Ax + By = CGraphing Ax + By = C

Example 2

Plot and label the graph of the equation y = –2x – 4.

Solution follows…

Topic4.2.2

Solution

Choose two values of x, then draw a table to help you find the y-values:

(2, –8)y = –2x – 4 = –2(2) – 4 = –82

(–2, 0)y = –2x – 4 = –2(–2) – 4 = 0–2

(x, y)yx

Solution continues…

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(0, –4)

Plot and label the graph of the equation y = –2x – 4.

Graphing Ax + By = CGraphing Ax + By = C

Example 2

Use the points in the table to plot the graph.

Solution (continued)

Topic4.2.2

x = 0

Check:

(0, –4) lies on the line — which means the line is correct.

(2, –8)

(–2, 0)

(x, y)

y = –2x – 4 = –2(0) – 4

= 0 – 4 = – 4

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Graphing Ax + By = CGraphing Ax + By = C

Guided Practice

Solution follows…

1. (–1, –3) and (3, 5)

2. (–3, 4) and (4, –3)

Graph the line through the two points in each of Exercises 1–2.

Topic4.2.2

(–1, –3)

(3, 5)

(–3, 4)

(4, –3)

2 1

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Graph and label the lines of the equations in Exercises 3–4.

Graphing Ax + By = CGraphing Ax + By = C

Guided Practice

Solution follows…

3. –x – 2y = 4

4. 2x – 3y = 6

Topic4.2.2

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2x – 3y = 6

–x – 2y = 4

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Graph and label the lines of the equations in Exercises 5–6.

Graphing Ax + By = CGraphing Ax + By = C

Guided Practice

Solution follows…

5. 5y – 3x = 15

6. 7y – 2x = –14

Topic4.2.2

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5y – 3x = 15

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7y – 2x = –14

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Independent Practice

Solution follows…

In Exercises 1–4, graph the line through each set of points.

Graphing Ax + By = CGraphing Ax + By = CTopic4.2.2

1. (–1, –2) and (2, 4)

2. (–1, –1) and (1, 3)

3. (0, 0) and (2, 6)

4. (0, –2) and (1, 1)

(2, 4)

(–1, –2)

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(1, 3)

(–1, –1)

23

(2, 6)

(0, 0)

4

(1, 1)

(0, –2)

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Independent Practice

Solution follows…

Graph and label the lines of the equations in Exercises 5–8.

Graphing Ax + By = CGraphing Ax + By = CTopic4.2.2

5. x + y = 8

6. y – x = 10

7. 2x + y = –3

8. 5x + y = –12 5

x + y = 8

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y – x = 10

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2x + y = –3

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5x + y = –12

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Independent Practice

Solution follows…

Graphing Ax + By = CGraphing Ax + By = CTopic4.2.2

9. –3x + y = –6

10. –10x + y = 21

11. 2x – y = –14

12. 6x + 2y = 18

Graph and label the lines of the equations in Exercises 9–12.

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–3x + y = –6

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–10x + y = 21

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2x – y = –14

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6x + 2y = 18

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Independent Practice

Solution follows…

Graphing Ax + By = CGraphing Ax + By = CTopic4.2.2

13. 8x + 4y = 24

14. 12x – 4y = 8

15. 3x – 9y = –27

16. 2x – 8y = 16

Graph and label the lines of the equations in Exercises 13–16.

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8x + 4y = 24

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12x – 4y = 8

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3x – 9y = –27

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2x – 8y = 16

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Round UpRound Up

It’s easy to make a mistake when working out y-values, so choose x-values that will make the algebra easy (for example, 0 and 1).

And it’s always a good idea to check your line by plotting a third point.

Graphing Ax + By = CGraphing Ax + By = CTopic4.2.2