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Exercises: Creating and Graphing Two-Variable Equations

Show your work for each problem. When writing equations, define your variables. When graphing, label both axes with variable names and units.

Grade 9·22 problems·~35 min·Common Core Math - HS Algebra·standard·hsa-ced-a-2
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A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Which ordered pair is a solution to y=4x−3y = 4x - 3?

2.

A car travels at a constant speed of 55 miles per hour. Which equation gives the distance dd (in miles) after tt hours?

3.

A graph shows hours worked (hh) on the horizontal axis and total pay in dollars (PP) on the vertical axis. Which statement correctly explains the axis assignment?

B

Fluency Practice

1.

A phone plan charges a flat fee of $25 per month plus $0.10 per text message. Write the equation for monthly cost CC (in dollars) in terms of the number of texts nn, then find the monthly cost when n=80n = 80. Enter the cost in dollars.

2.

A tank holds 400 liters of water and drains at 15 liters per minute. The volume remaining after tt minutes is V=400−15tV = 400 - 15t. What is VV when t=12t = 12? Enter your answer in liters.

3.

A fruit stand sells pears for $1.50 each and bunches of grapes for $2.00 each. A customer spends exactly $12. Let pp = pears and gg = grape bunches. Which equation represents all valid spending combinations?

Line y = -2x + 8 from (0,8) to (4,0) with four labeled candidate points A through D
4.

The line y=−2x+8y = -2x + 8 is graphed on axes with xx from 0 to 4 and yy from 0 to 8. Which ordered pair lies on this line?

5.

A ball drops from rest and its height hh (feet) after tt seconds is h=100−16t2h = 100 - 16t^2. What type of equation is this and what shape is its graph?

C

Varied Practice

1.

A baker earns $8 per loaf of bread sold. Which statement correctly identifies the independent and dependent variables?

2.

A car starts with 15 gallons of fuel and uses 0.04 gallons per mile. The equation is G=15−0.04dG = 15 - 0.04d, where GG = gallons remaining and dd = miles driven.

Complete the axis labels for a correct graph:

  • Horizontal axis variable name:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
  • Horizontal axis units:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
  • Vertical axis variable name:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
  • Vertical axis units:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
horizontal axis variable:
horizontal axis units:
vertical axis variable:
vertical axis units:
3.

The equation C=5nC = 5n models the cost CC (in dollars) of buying nn notebooks at $5 each. The point (3,15)(3, 15) lies on the graph. What does this point represent in context?

Two graphs of V = 500 - 20t: Student A shows a connected line; Student B shows isolated dots only.
4.

Two students graph V=500−20tV = 500 - 20t for 0≤t≤250 \le t \le 25.

Student A draws a straight line from (0,500)(0, 500) to (25,0)(25, 0) with axes labeled "Time (minutes)" and "Volume (liters)."
Student B plots only the six points (0,500)(0, 500), (5,400)(5, 400), (10,300)(10, 300), (15,200)(15, 200), (20,100)(20, 100), (25,0)(25, 0) and leaves them unconnected.

Which graph is correct, and why?

5.

The equation C=1.75m+3.50C = 1.75m + 3.50 models taxi cost CC (dollars) for mm miles driven. Which statement about the graph is true?

D

Word Problems

1.

A 500-liter water tank drains at a constant rate of 20 liters per minute. Let tt = time in minutes and VV = volume remaining in liters.

1.

Write the two-variable equation: $V = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $- $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   ⋅t\cdot t

initial volume (liters):
drain rate (liters per minute):
2.

The point (15,200)(15, 200) lies on the graph of your equation. Enter the volume (in liters) remaining after 15 minutes.

2.

A student has $24 to spend on sandwiches ($6 each) and drinks ($3 each) and wants to spend the entire amount.

The equation 6s+3d=246s + 3d = 24 represents all valid combinations. If the student buys 2 sandwiches, how many drinks can they purchase? Enter the number of drinks.

Graph of h = -16t^2 + 48t, a downward parabola peaking at (1.5, 36) and landing at (3, 0)
3.

A ball is launched upward from ground level with an initial velocity of 48 ft/s. Its height hh (feet) after tt seconds is h=−16t2+48th = -16t^2 + 48t.

1.

Find the maximum height of the ball. Enter the height in feet.

2.

After how many seconds does the ball return to the ground? Enter the time in seconds.

E

Error Analysis

1.

Marco graphed E=8hE = 8h, where hh = hours worked and EE = total earnings (dollars). He placed EE on the horizontal axis and hh on the vertical axis. His graph shows a line through the origin.

What error did Marco make?

2.

Priya graphed C=1.75m+3.50C = 1.75m + 3.50 for 0≤m≤100 \le m \le 10. She plotted the points (0,3.50)(0, 3.50), (2,7.00)(2, 7.00), (4,10.50)(4, 10.50), (6,14.00)(6, 14.00), (8,17.50)(8, 17.50), (10,21.00)(10, 21.00) as isolated dots without connecting them. She also labeled both axes as "number" without units.

Priya made two errors. Which answer correctly identifies both?

F

Challenge / Extension

1.

A rectangle has a fixed perimeter of 24 cm. Let xx = the length of one side in centimeters. The adjacent side then has length (12−x)(12 - x) cm.

Write the equation for area AA in terms of xx, then find the maximum possible area. Enter the maximum area in square centimeters.

2.

A bacteria colony starts at 200 and doubles every hour: P=200⋅2tP = 200 \cdot 2^t.

(a) Explain why the graph of this equation is NOT a straight line.
(b) Each additional hour multiplies PP by what factor? Explain what this means in context.

0 of 22 answered