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Intercepts by Substitution

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Graphs of Linear Functions from Intercepts
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What are the intercepts of 4x+2y=8 ? How could you use the intercepts to quickly graph the function?

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Khan Academy X and Y Intercepts

Guidance

To graph a linear function, you need to plot only two points. These points can then be lined up with a straight edge and joined to graph the straight line. While any two points can be used to graph a linear function, two points in particular that can be used are the x -intercept and the y -intercept. Graphing a linear function by plotting the x- and y- intercepts is often referred to as the intercept method.

The x -intercept is where the graph crosses the x -axis. Its coordinates are (x, 0) . Because all x -intercepts have a y -coordinate equal to 0, you can find an x -intercept by substituting 0 for y in the equation and solving for x .

The y -intercept is where the graph crosses the y -axis. Its coordinates are (0, y) . Because all y -intercepts have a x -coordinate equal to 0, you can find an y -intercept by substituting 0 for x in the equation and solving for y .

Example A

Identify the x- and y -intercepts for each line.

(a) 2x+y-6=0

(b) \frac{1}{2}x-4y=4

Solution:

(a)

&\text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& 2x+y-6=0 && 2x+y-6=0\\& 2x+({\color{red}0})-6=0 && 2({\color{red}0})+y-6=0\\& 2x-6=0 && y-6=0\\& 2x-6+6=0+6 && y-6+6=0+6\\& 2x=6 && y=6\\& \frac{2x}{2}=\frac{6}{2} && \text{The} \ y \text{-intercept is} \ (0, 6)\\& x=3\\& \text{The} \ x \text{-intercept is} \ (3, 0)

(b)

& \text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& \frac{1}{2}x-4y=4 && \frac{1}{2}x-4y=4\\& \frac{1}{2}x-4({\color{red}0})=4 && \frac{1}{2}({\color{red}0})-4y=4\\& \frac{1}{2}x-0=4 && 0-4y=4\\& \frac{1}{2}x=4 && -4y=4\\& \overset{1}{\cancel{2}}\left(\frac{1}{\cancel{2}}\right)x=2(4) && \frac{-4y}{-4}=\frac{4}{-4}\\& x=8 && y=-1\\& \text{The} \ x \text{-intercept is} \ (8, 0) && \text{The} \ y \text{-intercept is} \ (0, -1)

Example B

Use the intercept method to graph 2x-3y=-12 .

Solution:

& \text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& 2x-3y=-12 && 2x-3y=-12\\& 2x-3({\color{red}0})=-12 && 2({\color{red}0})-3y=-12\\& 2x-0=-12 && 0-3y=-12\\& 2x=-12 && -3y=-12\\& \frac{2x}{2}=\frac{-12}{2} && \frac{-3y}{-3}=\frac{-12}{-3}\\& x=-6 && y=4\\& \text{The} \ x \text{-intercept is} \ (-6, 0) && \text{The} \ y \text{-intercept is} \ (0, 4)

Example C

Use the x- and y -intercepts of the graph to identify the linear function that matches the graph.

a) y=2x-8

b) x-2y+8=0

c) 2x+y-8=0

The x -intercept is (–8, 0) and the y -intercept is (0, 4).

Solution: Find the x and y intercepts for each equation and see which matches the graph.

a) x intercept: 0=2x-8 \rightarrow x=4

y intercept: y=2(0)-8 \rightarrow y=-8

b) x intercept: x-2(0)+8=0 \rightarrow x=-8

y intercept: 0-2y+8=0 \rightarrow y=4

c) x intercept: 2x+0-8=0 \rightarrow x=4

y intercept: 2(0)+y-8=0 \rightarrow y=8

The x and y intercepts match for x-2y+8=0 so this is the equation of the line.

Concept Problem Revisited

The linear function 4x+2y=8 can be graphed by using the intercept method.

& \text{To determine the }x \text{-intercept, let } y=0. && \text{To determine the } y \text{-intercept, let } x=0.\\& \text{Solve for} \ `x\text{'}. && \text{Solve for} \ `y\text{'} .\\& 4x+2y=8 && 4x+2y=8\\& 4x+2({\color{red}0})=8 && 4({\color{red}0})+2y=8\\& 4x+{\color{red}0}=8 && {\color{red}0}+2y=8\\   & 4x=8 && 2y=8\\& \frac{4x}{4}=\frac{8}{4} && \frac{2y}{2}=\frac{8}{2}\\& x=2 && y=4\\& \text{The} \ x \text{-intercept is} \ (2, 0) && \text{The} \ y \text{-intercept is} \ (0, 4)

Plot the x -intercept on the x -axis and the y -intercept on the y -axis. Join the two points with a straight line.

Vocabulary

Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the x- and y -intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and joining them with a straight line.
x -intercept
An x -intercept of a relation is the x- coordinate of the point where the relation intersects the x -axis.
y -intercept
A y -intercept of a relation is the y- coordinate of the point where the relation intersects the y -axis.

Guided Practice

1. Identify the x- and y -intercepts of the following linear functions:

(i) 2(x-3)=y+4
(ii) 3x+\frac{2}{3}y-3=0

2. Use the intercept method to graph the following relation:

(i) 5x+2y=-10

3. Use the x- and y -intercepts of the graph, to match the graph to its function.

(i) 2x+y=6
(ii) 4x-3y-12=0
(iii) 5x+3y=15

Answers:

1. (i)

2(x-3)&=y+4 && \text{Simplify the equation}\\2(x-3)&=y+4\\2x-6&=y+4\\2x-6+6&=y+4+6\\2x&=y+10 && \text{You may leave the function in this form.}\\2x-y&=y-y+10\\2x-y&=10

If you prefer to have both variables on the same side of the equation, this form may also be used. The choice is your preference.

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 2x-y=10 && 2x-y=10\\& 2x-({\color{red}0})=10 && 2({\color{red}0})-y=10\\& 2x=10 && 0-y=10\\& \frac{2x}{2}=\frac{10}{2} && \frac{-y}{-1}=\frac{10}{-1}\\& x=5 && y=-10\\& \text{The} \ x \text{-intercept is} \ (5, 0) && \text{The} \ y \text{-intercept is} \ (0, -10)

(ii)

3x+\frac{2}{3}y-3&=0 && \text{Simplify the equation.}\\3(3x)+3\left(\frac{2}{3}\right)y-3(3)&=3(0) && \text{Multiply each term by 3.}\\3(3x)+\cancel{3}\left(\frac{2}{\cancel{3}}\right)y-3(3)&=3(0)\\9x+2y-9&=0\\9x+2y-9+9&=0+9\\9x+2y&=9

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 9x+2y=9 && 9x+2y=9\\& 9x+2({\color{red}0})=9 && 9({\color{red}0})+2y=9\\& 9x+0=9 && 0+2y=9\\& \frac{9x}{9}=\frac{9}{9} && \frac{2y}{2}=\frac{9}{2}\\& x=1 && y=4.5\\& \text{The} \ x \text{-intercept is} \ (1, 0) && \text{The} \ y \text{-intercept is} \ (0, 4.5)

2.

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 5x+2y=-10 && 5x+2y=-10\\& 5x+2({\color{red}0})=-10 && 5({\color{red}0})+2y=-10\\& 5x+0=-10 && 0+2y=-10\\& \frac{5x}{5}=\frac{-10}{5} && \frac{2y}{2}=\frac{-10}{2}\\& x=-2 && y=-5\\& \text{The} \ x \text{-intercept is} \ (-2, 0) && \text{The} \ y \text{-intercept is} \ (0, -5)

3. Identify the x- and y -intercepts from the graph.

The x -intercept is (3, 0)

The y -intercept is (0, -4)

Determine the x- and y -intercept for each of the functions. If the intercepts match those of the graph, then the linear function will be the one that matches the graph.

(i)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 2x+y=6 && 2x+y=6\\& 2x+({\color{red}0})=6 && 2({\color{red}0})+y=6\\& 2x=6 && 0+y=6\\& \frac{2x}{2}=\frac{6}{2} && y=6\\& x=3\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, 6)\\& \text{This matches the graph.} && \text{This does not match the graph.}

2x+y=6 is not the linear function for the graph.

(ii)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 4x-3y-12=0 && 4x-3y-12=0\\& 4x-3y-12+12=0+12 && 4x-3y-12+12=0+12\\& 4x-3y=12 && 4x-3y=12\\& 4x-3({\color{red}0})=12 && 4({\color{red}0})-3y=12\\& 4x-0=12 && 0-3y=12\\& 4x=12 && -3y=12\\& \frac{4x}{4}=\frac{12}{4} && \frac{-3y}{-3}=\frac{12}{-3}\\& x=3 && y=-4\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, -4)\\& \text{This matches the graph.} && \text{This matches the graph.}

4x-3y-12=0 is the linear function for the graph.

(iii)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 5x+3y=15 && 5x+3y=15\\& 5x+3({\color{red}0})=15 && 5({\color{red}0})+3y=15\\& 5x+0=15 && 0+3y=15\\& 5x=15 && 3y=15\\& \frac{5x}{5}=\frac{15}{5} && \frac{3y}{3}=\frac{15}{3}\\& x=3 && y=5\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, 5)\\& \text{This matches the graph.} && \text{This does not match the graph.}

5x+3y=15 is not the linear function for the graph.

Practice

For 1-10, complete the following table:

Function x -intercept y -intercept
7x-3y=21 1. 2.
8x-3y+24=0 3. 4.
\frac{x}{4}-\frac{y}{2}=3 5. 6.
7x+2y-14=0 7. 8.
\frac{2}{3}x-\frac{1}{4}y=-2 9. 10.

Use the intercept method to graph each of the linear functions in the above table.

  1. 7x-3y=21
  2. 8x-3y+24=0
  3. \frac{x}{4}-\frac{y}{2}=3
  4. 7x+2y-14=0
  5. \frac{2}{3}x-\frac{1}{4}y=-2

Use the x- and y -intercepts to match each graph to its function.

a. 7x+5y-35=0
b. y=5x+10
c. 2x+4y+8=0
d. 2x+y=2
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