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Algebra 1

1.11.2 Relating Two-Variable Equations, Their Graphs, and Situations

Algebra 11.11.2 Relating Two-Variable Equations, Their Graphs, and Situations

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Activity

For this activity, you will use two graphs that represent situations you saw in other activities.

For questions 1 – 3, use the graph below: The graph represents a = 450 20 t a = 450 20 t , which describes the relationship between gallons of water in a tank and time in minutes.

Line graph showing a decrease from 500 to 0 gallons in a tank over 24 minutes. The x-axis is labeled time (minutes) from 0 to 26, and the y-axis is labeled amount in tank (gallons) from 0 to 500.
1.

Where on the graph can we see the 450?

2.

Where can we see the -20?

3.

What do the numbers 450 and -20 mean in the situation that is graphed?

4.

The graph represents 6 x + 9 y = 75 6 x + 9 y = 75 . It describes the relationship between pounds of almonds and figs and the dollar amount Clare spent on them. Suppose a classmate says, “I am not sure the graph represents 6 x + 9 y = 75 6 x + 9 y = 75 because I don’t see the 6, 9, or 75 on the graph.” How would you show your classmate that the graph indeed represents this equation?

A graph with dried figs (pounds) on the y-axis and almonds (pounds) on the x-axis shows a straight, downward-sloping line. The grid is marked from 0 to 14 on the x-axis and 0 to 10 on the y-axis.

Self Check

For the line graphed below, what is the y -intercepts?


  1. ( 5 , 0 )
  2. ( 0 , 5 )
  3. ( 5 , 0 )
  4. ( 0 , 5 )

Additional Resources

Finding Intercepts from Graphs and Equations

The points where a line crosses the x x -axis and the y y -axis are called the intercepts of the line.

Let’s look at the graphs of the lines.

Four coordinate plane graphs, each showing a different linear equation with its line and two labeled intersection points. Each graph is labeled (a) 2x+y=6, (b) 3x-4y=12, (c) x-y=5, (d) y=-2x.

First, notice where each of these lines crosses the x x -axis.

Now, let’s look at the points where these lines cross the y y -axis.

Figure The Line Crosses the x x -axis at: Ordered Pair for this Point The Line Crosses the y y -axis at: Ordered Pair for this Point
Figure (a) 3 ( 3 , 0 ) ( 3 , 0 ) 6 ( 0 , 6 ) ( 0 , 6 )
Figure (b) 4 ( 4 , 0 ) ( 4 , 0 ) -3 ( 0 , 3 ) ( 0 , 3 )
Figure (c) 5 ( 5 , 0 ) ( 5 , 0 ) -5 ( 0 , 5 ) ( 0 , 5 )
Figure (d) 0 ( 0 , 0 ) ( 0 , 0 ) 0 ( 0 , 0 ) ( 0 , 0 )
General Figure a a ( a , 0 ) a , 0 ) b b (0, b ) b )

X X -INTERCEPT and Y Y -INTERCEPT of a Line

The x x -intercept is the point ( a , 0 ) ( a , 0 ) where the line crosses the x x -axis.

The y y -intercept is the point ( 0 , b ) ( 0 , b ) where the line crosses the y y -axis.

The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b.

Find the x x - and y y -intercepts from the Equation of a Line

Use the equation of the line. To find:

  • the x x -intercept of the line, let y = 0 y = 0 and solve for x x .
  • the y y -intercept of the line, let x = 0 x = 0 and solve for y y .

Example

Find the intercepts of 2 x + y = 8 2 x + y = 8

Solution

To find the x x -intercept:

Step 1 - Let y = 0 y = 0 .

2 x + 0 = 8 2 x + 0 = 8

Step 2 - Solve for x x .

x = 4 x = 4

Step 3 - Write the intercept as a point.

( 4 , 0 ) ( 4 , 0 )

To find the y y -intercepts:

Step 1 - Let x = 0 x = 0 .

2 ( 0 ) + y = 8 2 ( 0 ) + y = 8

Step 2 - Solve for y y .

y = 8 y = 8

Step 3 - Write the intercept as a point.

( 0 , 8 ) ( 0 , 8 )

Try it

Try It: Finding Intercepts from Graphs and Equations

Find the x x - and y y -intercept of x + 4 y = 8 x + 4 y = 8 .

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