Refer to our Texas Go Math Grade 7 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 7 Lesson 2.3 Answer Key Proportional Relationships and Graphs.

## Texas Go Math Grade 7 Lesson 2.3 Answer Key Proportional Relationships and Graphs

**Your Turn**

Question 1.

Jared rents bowling shoes for $6 and pays $5 per bowling game. Graph the data. Is the relationship a proportional relationship? Explain.

Answer:

The relationship isn’t proportional, because the points form a line that does not go through the origin.

**Example**

The graph shows the relationship between time in minutes and the number of miles Damon runs. Write an equation for this relationship.

STEP 1: Choose a point on the graph and tell what the point represents.

The point (25, 2.5) represents the distance (2.5 miles) that Damon runs in 25 minutes.

STEP 2: What ¡s the constant of proportionality?

Because distance \(\frac{\text { distance }}{\text { time }}=\frac{2.5 \mathrm{mi}}{25 \mathrm{~min}}=\frac{1}{10}\), the constant of proportionality is \(\frac{1}{10}\).

STEP 3: Write an equation in the form y = kx.

y = \(\frac{1}{10}\)x

**Reflect**

Question 2.

Communicate Mathematical Ideas What does the point (0, 0) on the graph represent?

Answer:

The point (0, 0) represents a start on the graph, origin.

Question 3.

What If? Esther runs faster than Damon. Suppose you drew a graph representing the relationship between time in minutes and distance run for Esther. How would the graph compare to the one for Damon?

Answer:

Esther’s Line wou[d be steeper than Damon’s

**Your Turn**

Question 4.

The graph shows the relationship between the distance a bicyclist travels and the time in hours.

a. What does the point (4, 60) represent?

Answer:

The point (4,60) represents the distance (60 miles) that the bicyclist travels in 4 hours

b. What is the constant of proportionality?

Answer:

The constant is equal to 15 because \(\frac{\text { distance }}{\text { time }}=\frac{60}{4}\) = 15. Only one point is enough since the relationship is proportional.

c. Write an equation in the form y = kx for this relationship.

Answer:

The equation is equal to y = 15x.

**Texas Go Math Grade 7 Lesson 2.3 Guided Practice Answer Key**

**Complete each table. Tell whether the relationship is a proportional relationship. Explain why or why not. xpIore Activity)**

Question 1.

A student reads 65 pages per hour.

Answer:

3 × 65 = 195

5 × 65 = 325

585 ÷ 65 = 9

10 × 65 = 650

The relationship is proportional because we made sure with our calculations that all constants are equal to 65.

Question 2.

A babysitter makes $7.50 per hour.

Answer:

2 × 7.51 = 15

22.50 ÷ 7.50 = 3

5 × 7.50 = 37.5()

60 ÷ 7.50 = 8

The relationship is proportional because we made sure with our calculations that all constants are equal to 7.50.

**Tell whether the relationship is a proportional relationship. Explain why or why not. (Explore Activity and Example 1)**

Question 3.

Answer:

The relationship is not proportional because the line does not go through the origin.

Question 4.

Answer:

The relationship is proportional, the constant is equal to 2 because \(\frac{2}{1}\) = \(\frac{4}{2}\) = \(\frac{10}{5}\) = \(\frac{16}{8}\) = 2. The equation is equal to y = 2x.

**Write an equation of the form y = kx for the relationship shown in each graph.**

Question 5.

Answer:

The relationship is proportional because points form a line through the origin.

Thus, we need only one point to determine the constant

7 ÷ 2 = 3.5

y = Balloon height(ft)

x = Time(s)

Equation: y = 3.5x

Question 6.

Answer:

The relationship is proportional because points form a line through the origin.

Thus, we need only one point to determine the constant

2 ÷ 8 = 025

y = Cost ($)

x = Number of items

Equation: y = 0.5x

**Essential Question Check-In**

Question 7.

How does a graph show a proportional relationship?

Answer:

The graph forms a line which passes through the origin.

**Texas Go Math Grade 7 Lesson 2.3 Independent Practice Answer Key**

**For Exercises 8-12, the graph shows the relationship between time and distance run by two horses.**

Question 8.

Explain the meaning of the point (0, 0).

Answer:

The point (0,0) represents start, position of horses right before they started to run.

Question 9.

How long does it take each horse to run a mile?

Answer:

We can see from the graph that horse A takes 4 minutes to run a mile, while horse B takes 2.5 minutes to run a mile.

Question 10.

**Multiple Representations** Write an equation for the relationship between time and distance for each horse.

Answer:

Horse A runs 1 mile in 4 minutes. That means he runs \(\frac{1}{4}\) miles per minute.

Horse B runs 1 mile in 2.5 minutes That means he runs \(\frac{1 \times 2}{2.5 \times 2}\) = \(\frac{2}{5}\) miles per minute.

x = Time (min)

y = Distance(miles)

Horse A: y = \(\frac{1}{4}\) x

Horse B: y = \(\frac{2}{5}\) x

Question 11.

Draw Conclusions At the given rates, how far would each horse run in 12 minutes?

Answer:

Use the equations from 11:

Horse A:

y = \(\frac{1}{4}\) x

y = \(\frac{1}{4}\)(12)^{3}

y = 3

Horse B:

y = \(\frac{2}{5}\) x

y = \(\frac{2}{5}\) (12)

y = \(\frac{25}{4}\)

y = 4\(\frac{4}{5}\)

Horse A passes 3 miles in 12 minutes.

Horse B passes 4\(\frac{4}{5}\) in 12 minutes.

Question 12.

**Analyze Relationships** Draw a line on the graph representing a horse than runs faster than horses A and B.

Answer:

Question 13.

A bullet train can travel at 170 miles per hour. Will a graph representing distance in miles compared to time in hours show a proportional relationship? Explain.

Answer:

The graph will show a proportional relationship because of the constant unit rate, 170 miles per hour.

Question 14.

Critical Thinking When would it be more useful to represent a proportional relationship with a graph rather than an equation?

Answer:

It would be easier to draw graphs when we have whole numbers

It would not be easy to draw a graph if a constant is a long decimal number, or a fraction with big numbers. Thus, we rather use equation in this case.

Question 15.

Multiple Representations Bargain DVDs cost $5 each at Mega Movie.

a. Graph the proportional relationship that gives the cost y in dollars of buying x bargain DVDs.

Answer:

Graph

b. Give an ordered pair on the graph and explain its meaning in the real world context.

Answer:

The point (4, 20) represents $20 you have to pay for renting 4 DVDs.

**The graph shows the relationship between distance and time as Glenda swims.**

Question 16.

How far did Glenda swim in 4 seconds?

Answer:

Glenda swam 8 feet in 4 seconds.

Question 17.

**Communicate Mathematical Ideas** Is this a proportional relationship? Explain your reasoning.

Answer:

This is a proportional relationship because points form a line through the origin.

Question 18.

**Multiple Representations** Write an equation that shows the relationship between time and distance. ________________________________

Answer:

Because the relationship is proportional we can calculate the constant k. From point (2, 4) we conclude the constant is equal to 2.

x = Time(s)

y = Distance(ft)

Thus, the equation is y = 2x.

**H.O.T.S Focus On Higher Order Thinking**

Question 19.

**Make a Conjecture** If you know that a relationship is proportional and are given one ordered pair other than (0, 0), how can you find another pair?

Answer:

From the point that is given to us we can draw a line on the graph through that point and (0, 0). Then we can find which ever point we need.

**The tables show the distance traveled by three cars.**

Question 20.

**Communicate Mathematical Ideas** Which car is not traveling at a constant speed? Explain your reasoning.

Answer:

Car 3 is not traveling at a constant speed because \(\frac{65}{1}\) ≠ \(\frac{85}{2}\).

Question 21.

**Make a Conjecture** Car 4 is traveling at twice the rate of speed of car 2. How will the table values for car 4 compare to the table values for car 2?

Answer:

The time column will stay the same but the distance column will double its values because the constant doubles.