Describing Relationships Using Words and Equations

1.3.2 Describing Relationships Using Words and Equations

1.3.2 • Describing Relationships Using Words and Equations

Activity

Create a math story that describes how the two quantities in each table are related.

Table A

Number of Laps, $x$	0	1	2.5	6	9
Meters Run, $y$	0	400	1,000	2,400	3,600

This table shows the relationship between the number of laps run (x) and the total distance covered in meters (y), demonstrating that 1 lap equals 400 meters.

Your answer may vary, but here is an example

Every lap is 400 meters, so the distance run is 400 times the number of laps.

Table B

Meters From Home, $x$	0	75	128	319	396
Meters From School, $y$	400	325	272	81	4

Distances from home (x) and school (y) are presented for five sequential observations, illustrating how these distances change relative to each other.

Your answer may vary, but here is an example.

The distance between home and school is 400 meters. Every meter traveled away from home is a meter closer to school.

Table C

Electricity Bills in Dollars, $x$	85	124	309	816
Total Expenses in Dollars, $y$	485	524	709	1,216

This table displays four pairs of electricity bills (x) and total expenses (y) in dollars, showing how these financial metrics relate.

Your answer may vary, but here is an example.

Total expenses can be found by adding $400 to electricity bills. (The $400 could represent rent payment.)

Table D

Monthly Salary in Dollars, $x$	872	998	1,015	2,110
Amount Deposited in Dollars, $y$	472	598	615	1,710

This table displays four paired observations of monthly salaries (x) and corresponding amounts deposited (y) in dollars.

Your answer may vary, but here is an example.

The amount deposited is $400 less than the monthly salary. (The employer might deduct $400 per paycheck for taxes, health insurance, retirement savings, and so on. Or, the employee might first pay $400 in rent or other expenses before depositing the rest of the monthly salary.)

For questions 5 – 8, match each table from 1 - 4 to an equation that represents the relationship.

Equation 1: $400 + x = y$

Compare your answer:

Table C

Equation 2: $x - 400 = y$

Compare your answer:

Table D

Equation 3: $x + y = 400$

Compare your answer:

Table B

Equation 4: $400 \cdot x = y$

Compare your answer:

Table A

Are you ready for more?

Extending Your Thinking

Express every number between 1 and 20 at least one way using exactly four 4s and any operations. For example, 1 could be written as $\frac{4}{4} + 4 - 4$ .

Compare your answer:

Your answer may vary, but here are some samples:

$1 = \frac{4}{4} + 4 - 4$

$2 = \frac{4}{4} + \frac{4}{4}$

$3 = \sqrt{4} + \sqrt{4} - \frac{4}{4}$

$4 = \sqrt{4} + \sqrt{4} + 4 - 4$

$5 = \sqrt{4} + \sqrt{4} + \frac{4}{4}$

$6 = 4 + \frac{4 + 4}{4}$

$7 = 4 + 4 - \frac{4}{4}$

$8 = \sqrt{4} + \sqrt{4} + \sqrt{4} + \sqrt{4}$

$9 = 4 + 4 + \frac{4}{4}$

$10 = (4 + \frac{4}{4}) \sqrt{4}$

$11 = \frac{4! \sqrt{4} - 4}{4}$

$12 = 4 (4 - \frac{4}{4})$

$13 = \frac{4! \sqrt{4} + 4}{4}$

$14 = 4 \cdot 4 - \frac{4}{\sqrt{4}}$

$15 = 4 \cdot 4 - \frac{4}{4}$

$16 = 4 + 4 + 4 + 4$

$17 = 4 \cdot 4 + \frac{4}{4}$

$18 = 4 \cdot 4 + 4 - \sqrt{4}$

$19 = 4! - (4 + \frac{4}{4})$

$20 = 4 (4 + \frac{4}{4})$

Video: Describing Relationships

Watch the following video to learn more about describing relationships using words and equations, specifically looking at the base and height of different rectangles.

Access multimedia content

Self Check

Miles from Home, $x$	0	5	10
Miles from School, $y$	30	25	20

Using the table above, what is an equation that shows the relationship between miles from home and miles from school?

$y=30x$
$y=30-x$
$y=20+x$
$y=30+x$

Additional Resources

Modeling Linear Equations

Given a real-world problem, model a linear equation to fit it using these steps:

Step 1 - Read the problem.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.

Step 3 - Write a sentence or create a table using the relationship among the values.

Step 4 - Translate the sentence or table into an equation

Let’s look at an example. Using the steps above, we can model a linear equation to solve a real-world application.

Example

Cell phone Company A charges a monthly service fee of $34 plus $0.05/minute talk time.

Step 1 - Read the problem.
Cell phone Company A charges a monthly service fee of $34 plus $0.05/minute talk time.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.
$t$ : total cost (in dollars)
$m$ : number of minutes

Step 3 - Write a sentence or create a table using the relationship among the values.

Minutes Talked in One Month	0	1	2	3	100
Total for One Month (Dollars)	34	34.05	34.10	34.15	39

Monthly phone bill total based on minutes talked.

Company A: The total cost, $t$ , equals the fee, 34, plus 0.05 per minute, $0.05 m$ .

Step 4 - Translate the sentence or table into an equation.
$t = 34 + 0.05 m$

Try it

Modeling Linear Equations

Cell phone Company B charges a monthly service fee of $40 plus $0.04/min talk-time. Write an equation to describe the relationship between the number of minutes and the total cost.

Compare your answer:

Step 1 - Read the problem.

Cell phone Company B charges a monthly service fee of $40 plus $0.04/minute talk time. Write an equation to describe the relationship between the number of minutes and the total cost.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.

Monthly Fee: $40, Cost per Minute: $0.04

$t$ : total cost (in dollars) $m$ : number of minutes

Step 3 - Write a sentence or create a table using the relationship among the values.

Company B:

Minutes Talked in One Month	0	1	2	3	100
Total for One Month (Dollars)	40	40.04	40.08	40.12	44

This table illustrates the total monthly cost based on the number of minutes talked, showing a base fee and a per-minute charge for various usage scenarios.

The total cost, $t$ , equals the fee, 40, plus 0.04 per minute.

Step 4 - Translate the sentence or table into an equation. $t = 40 + 0.04 m$