Relating Two-Variable Equations, Their Graphs, and Situations

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

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

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 $a = 450 - 20 t$ , 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.

Where on the graph can we see the 450?

Compare your answer:

450 is where the graph intersects the vertical axis.

Where can we see the −20?

Compare your answer:

−20 is the slope.

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

Compare your answer:

Your answer may vary, but here is a sample. 450 is the gallons of water in the tank at 0 minutes, before the tank starts to be drained. −20 means the gallons of water (vertical value) drops by 20 for every 1 minute increase in time (horizontal value).

The graph represents $6 x + 9 y = 75$ . 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 $6 x + 9 y = 75$ 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.

Compare your answer:

Your answer may vary, but here is a sample.

If we substitute 0 for $x$ in the equation and solve for $y$ , we get $\frac{75}{9}$ or $8 \frac{1}{3}$ . That combination is the point $(0, 8 \frac{1}{3})$ or the $y$ -intercept of the graph. If we substitute 0 for $y$ in the equation and solve for $x$ , we get 12.5 for $x$ . That combination is the point (12.5,0) or the $x$ -intercept of the graph.
If we substitute the $x$ - and $y$ -values of any point on the graph in the equation, the equation remains true.
If we rewrite the equation and solve for $y$ , we have $y = \frac{75 - 6 x}{9}$ or $y = 8 \frac{1}{3} - \frac{2}{3} x$ . The $8 \frac{1}{3}$ matches where the graph intersects the $y$ -axis and $- \frac{2}{3}$ matches the slope of the graph. (For every 3 additional pounds of almonds that Clare bought, she could buy 2 fewer pounds of figs.)
From the graph, we can see that the $y$ -intercept is $(0, 8 \frac{1}{3})$ and the slope is $- \frac{2}{3}$ , so the equation of the line is $y = 8 \frac{1}{3} - \frac{2}{3} x$ . Multiplying each side of the equation by 9 (an acceptable move) gives an equivalent equation, $9 y = 75 - 6 x$ , which can be rewritten as $6 x + 9 y = 75$ .

Self Check

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

$(5, 0)$
$(0, 5)$
$(-5,0)$
$(0, -5)$

Additional Resources

Finding Intercepts from Graphs and Equations

The points where a line crosses the $x$ -axis and the $y$ -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 $x$ -axis.

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

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

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

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

The $y$ -intercept is the point $(0, b)$ where the line crosses the $y$ -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 $x$ - and $y$ -intercepts from the Equation of a Line

Use the equation of the line. To find:

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

Example

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

Solution

To find the $x$ -intercept:

Step 1 - Let $y = 0$ .

$2 x + 0 = 8$

Step 2 - Solve for $x$ .

$x = 4$

Step 3 - Write the intercept as a point.

$(4, 0)$

To find the $y$ -intercepts:

Step 1 - Let $x = 0$ .

$2 (0) + y = 8$

Step 2 - Solve for $y$ .

$y = 8$

Step 3 - Write the intercept as a point.

$(0, 8)$

Try it

Finding Intercepts from Graphs and Equations

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

Compare your answer: (8, 0) (0, 2)

Here is how to find the intercepts using a general strategy.

To find the $x$ -intercept,

Step 1 Let $y = 0$ .

$x + 4 (0) = 8$

Step 2 - Solve for $x$ .

$x = 8$

Step 3 - Write the intercept as a point.

$(8, 0)$

To find the $y$ -intercept:

Step 1 - let $x = 0$

$(0) + 4 y = 8$

Step 2 - Solve for $y$ .

$y = 2$

Step 3 - Write the intercept as a point.

$(0, 2)$

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

Activity

Self Check

Additional Resources

Finding Intercepts from Graphs and Equations

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

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

Try it

Finding Intercepts from Graphs and Equations

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

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