Finding x- and y-Intercepts of a Linear Equation

The Lesson Activities will help you meet these educational goals:

  • Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively, and use mathematics to model real-world situations.
  • Inquiry—You will analyze results and draw conclusions.
  • 21stCentury Skills—You will use critical-thinking and problem-solving skills.

Directions

Save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Finding the Intercepts of Linear Equations
  1. Many people enjoy challenging themselves by running in marathons. A marathon is a 26-mile race. The runners wear numbers that identify them so their progress can be tracked at checkpoints throughout the race. Such races can be modeled using equations.

An athlete is running in a marathon that eventually finishes at a local high school. After x minutes, the distance the athlete is from the school is given by y (in miles), where x+10y = 260. (Note: 0 ≤ x ≤ 10 and 0 ≤ y ≤ 26.)

  1. Use the relationship between x and y to find the time it takes for the runner to reach the school.

Sample answer:

The value of y at the school (finish line) will be 0.

Substitute 0 = y in the equation:

  1. Use the relationship between x and y to find the distance between the runner and the high school at the start line.

Sample answer:

The value of x at the start line will be 0.

Substitute 0 = x in the equation:

  1. If the runner ran for 40 minutes from the start line, what is the remaining distance between the runner and the school?

Sample answer:

x + 10y = 260

x = 40 minutes

  1. How much time will the runner take to complete half the distance between the start line and the high school?

Sample answer:

Since the entire distance is 26 miles, the value of y halfway between the start line and the school is

The runner will complete half the distance between the start line and the school in 130 minutes.

  1. What do the x- and y-intercepts represent in this scenario?

Sample answer:

The x-intercept represents the time required to complete the marathon, and the y-intercept represents the total distance to be covered to complete the marathon.

  1. Jason works as an administrative assistant for a local lawyer's office. The office is running low on pens and notepads,so he is asked to buy $150 worth of pens and notepads. Pens cost $1, and notepads cost $2. If the number of pens he buys is represented by x and the number of notepads he buys is represented by y, the situation can be given by x + 2y = 150, where 0 ≤ x ≤ 150, 0 ≤ y ≤ 75 and x and y can take only integer values.
  1. Assuming that the stationery store is out of notepads, how many pens would Jason have to purchase to spend the entire amount he is given?

Sample answer:

The value of y if the store is out of notepads = 0.

Substitute 0 as the value for y in the equation:

Jason would have to purchase 150 pens to spend the entire amount on pens.

  1. Assuming that the stationery store is out of pens, how many notepads would Jason have to purchase to spend the entire amount he is given?

Sample answer:

Value of x if the store is out of pens = 0.

Substitute 0 as the value for x in the equation:

Jason would have to purchase 75 notepads to spend the entire amount on notepads.

  1. What do the x- and y-intercepts represent in this scenario?

Sample answer:

The x-intercept represents the maximum number of pens that can be bought for $150, and the y-intercept represents the maximum number of pens that can bought for $150.

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