Writing Systems of Equations from Tables

2.2.2 Writing Systems of Equations from Tables

2.2.2 • Writing Systems of Equations from Tables

Activity

This summer, the Farmer’s Almanac is calling for very high temperatures in central Texas. Mr. and Mrs. Flores were talking about ways to keep their four children cool during the summer and also ways that the family could have fun together. They decided that this summer they would purchase a family membership to the community pool.

Mrs. Flores called the community pool to find out different membership plans for a family of six. The receptionist at the community center told her that there are two options. Option A is to pay $100 for the summer and $20 per visit. The second option, Option B, is to pay $250 for the summer and $5 per visit.

At dinner, the Flores family talked about how many times a week they think they will visit the pool and how many times total in the summer they will visit.

How can the Flores family decide which option to choose? Which option do you think they will pick and why?

Make a table of values for Option A for 0–10 visits.

Option A:

0	1	2	3	4	5	6	7	8	9	10

Compare your answer:

0	1	2	3	4	5	6	7	8	9	10
100	120	140	160	180	200	220	240	260	280	300

Make a table of values for Option B for 0–10 visits.

Option B:

0	1	2	3	4	5	6	7	8	9	10

Compare your answer:

0	1	2	3	4	5	6	7	8	9	10
250	255	260	265	270	275	280	285	290	295	300

For Option A, identify the $y$ -intercept (where $x = 0$ ).

100

For Option A, identify the slope between any two points.

For Option A, write an equation of the line in slope-intercept form.

Compare your answer:

$y = 20 x + 100$

For Option B, identify the $y$ -intercept (where $x = 0$ ).

250

For Option B, identify the slope between any two points.

For Option B, write an equation of the line in slope-intercept form.

Compare your answer:

$y = 5 x + 250$

Write the system of equations.

Compare your answer:

Answer:

$y = 20 x + 100$

$y = 5 x + 250$

10.

Use the graphing tool or technology outside the course. Graph your system of equations from question 9 that represents this scenario using the Desmos tool below.

Access multimedia content

Compare your answer:

Graph comparing Option A and B with lines intersecting. Option A is rising steeply, shown in a dashed line, while Option B is a shallower solid line.

11.

What is the point of intersection of the two lines?

Compare your answer:

$(10, 300)$

12.

Talk with your partner. What does this intersection mean in the context of the problem?

Compare your answer:

At 10 visits, the cost is the same at $300.

13.

Write a paragraph to describe when Option A is better and when Option B is better. Share your paragraph with your partner. Read each other’s paragraphs and then discuss if you agree or disagree.

Compare your answer:

Your answer may vary, but here is a sample. The two options cost the same at 10 visits. Since Option A is less money up front, Option A will be less expensive until the $10^{th}$ visit. After 10 visits, however, Option B is less expensive.

Self Check

A streaming service offers two options for subscription plans. Option A charges $10.99 a month plus $2 per movie. Option B charges $5.99 a month plus $3 per movie. The table below describes this. Which of the following systems of equations represents the two options for one month?

$x$ (Number of movies	0	1	2	3
Option A	10.99	12.99	14.99	16.99
Option B	5.99	8.99	11.99	14.99

$y=2x +10.99$

$y=3x+5.99$
$y=3x +10.99$

$y=2x+5.99$
$y=2x -10.99$

$y=3x-5.99$

$y=10.99x+2$

$y=5.99x +3$

Additional Resources

Representing a Linear Function in Tabular Form

A system of equations is two or more equations. Often, each equation in a system of equations represents a situation. Like other equations, the equations that make up the system can be represented by a table. When writing any equation from a table into slope-intercept form, two things are needed: a slope and point, often the $y$ -intercept.

Example

Suppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. Here is a table that shows this constant speed.

Step 1 - Find the $y$ -intercept.

Notice that when $t = 0, D (t) = 250$ . This is the $y$ -intercept.

Step 2 - Find the slope.

To find the slope, find the change of $y$ over the change of $x : m = \frac{83}{1} = 83$ .

Step 3 - Write the equation in slope-intercept form.

$y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.

$y = 83 x + 250$
Try it

Representing a Linear Function in Tabular Form

A new plant food was introduced to a young tree to test its effect on the height of the tree. The table shows the height of the tree, in feet, $x$ months since the measurements began. Write a linear function, $H (x)$ , where $x$ is the number of months since the start of the experiment.

$x$ 0 2 4 8 12

$H (x)$ 12.5 13.5 14.5 16.5 18.5

This table displays various input values for 'x' and their corresponding output values for the function 'H(x)'.

Compare your answer:

Here is how to write an equation from the table.

Step 1 - Find the $y$ -intercept.

Notice that when $x = 0$ , $H (x) = 12.5$ . This is the $y$ -intercept.

Step 2 - Find the slope.

To find the slope, find the change of $y$ over the change of $x$ : $m = \frac{1}{2} = 0.5$ .

Step 3 - Write the equation in slope-intercept form.

$y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.

$y = 0.5 x + 12.5$

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