Linear Models

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Answer Key & Explanations

Linear Models · Grade 8 · Worksheet 3

1. Emma is training for a charity walk-a-thon. She tracks her total distance walked over several days and finds that the relationship between the number of days she has been training (x) and her total distance walked in miles (y) can be modeled by the linear equation y = 3.5x + 5. What does the number 3.5 represent in this context, and what does the number 5 represent? Answer: The slope 3.5 represents the number of miles Emma walks each day. The y-intercept 5 represents the total miles she walked before starting the training. Solution: The equation is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Here m = 3.5.
   Full step-by-step solution

   Step 1: The equation is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Step 2: Here m = 3.5. Since x is the number of days and y is total distance walked, the slope tells how much y increases for each increase of 1 in x. So, for each additional day of training, Emma walks 3.5 more miles. Thus, 3.5 represents the miles Emma walks per day. Step 3: The y-intercept b = 5. This is the value of y when x = 0. When x = 0 days of training, y = 5 miles. This means before training started, Emma had already walked 5 miles (perhaps from a previous walk or an initial distance). Final answer: The slope 3.5 means Emma walks 3.5 miles each day of training. The y-intercept 5 means she had already walked 5 miles before starting the training.

2. A city's public transportation system uses the equation R = 1.75d + 2.50 to calculate the fare R in dollars for a ride based on distance d in miles. If Emma pays $9.75 for her bus ride to school, how many miles did she travel? Answer: 4.14 Solution: Write down the given equation: R = 1.75d + 2.50 Substitute the known total fare: 9.75 = 1.75d + 2.50 Subtract the fixed fee from both sides: 9.75 - 2.50 = 1.75d Calculate: 7.25 = 1.75d Divide both sides by 1.75 to solve for d: d = 7.25 ÷ 1.75 Calculate the division: d = 4.142857...
   Full step-by-step solution

   Step 1: Write down the given equation: R = 1.75d + 2.50 Step 2: Substitute the known total fare: 9.75 = 1.75d + 2.50 Step 3: Subtract the fixed fee from both sides: 9.75 - 2.50 = 1.75d Step 4: Calculate: 7.25 = 1.75d Step 5: Divide both sides by 1.75 to solve for d: d = 7.25 ÷ 1.75 Step 6: Calculate the division: d = 4.142857... Step 7: Round to two decimal places: d = 4.14 Emma traveled approximately 4.14 miles.

3. Aroha is tracking the height of a plant over time. She graphs the data and draws a line of best fit. The line passes through the points (5, 13) and (11, 25), where x represents the number of weeks since planting and y represents the height in centimeters. Write the equation of the line in slope-intercept form and interpret the meaning of the slope and the y-intercept in the context of the plant's growth. Answer: y = 2x + 3; The slope of 2 means the plant grows 2 centimeters per week. The y-intercept of 3 means the plant was 3 centimeters tall when planted (at week 0). Solution: Find the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Use points (5, 13) and (11, 25): m = (25 - 13) / (11 - 5) = 12 / 6 = 2. Use the slope-intercept form y = mx + b.
   Full step-by-step solution

   Step 1: Find the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Use points (5, 13) and (11, 25): m = (25 - 13) / (11 - 5) = 12 / 6 = 2. So the slope is 2. Step 2: Use the slope-intercept form y = mx + b. Substitute one point, say (5, 13), and the slope: 13 = 2(5) + b, so 13 = 10 + b, b = 3. The equation is y = 2x + 3. Step 3: Interpret the slope (m = 2): For each additional week (increase in x by 1), the height increases by 2 cm. So the plant grows 2 cm per week. Step 4: Interpret the y-intercept (b = 3): When x = 0 (at the time of planting), the height is 3 cm. So the plant was 3 cm tall when planted. The answer is y = 2x + 3; slope means 2 cm per week growth, y-intercept means 3 cm initial height.

4. Noah is saving money to buy a new video game console. He already has $61 saved and plans to save $16 each week from his allowance. The total amount of money Noah has saved after w weeks can be modeled by the equation y = 16w + 61. In this equation, what does the number 16 represent in the context of Noah's savings? What does the number 61 represent? Answer: The number 16 represents the amount Noah saves each week ($16 per week). The number 61 represents the initial amount Noah already had saved ($61). Solution: Identify the form of the equation. The equation y = 16w + 61 is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Interpret the slope (m = 16).
   Full step-by-step solution

   Step 1: Identify the form of the equation. The equation y = 16w + 61 is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Step 2: Interpret the slope (m = 16). The slope represents the rate of change. In this context, w represents weeks and y represents total savings. So 16 means Noah saves $16 each week. Step 3: Interpret the y-intercept (b = 61). The y-intercept is the value of y when w = 0. This means before any weeks have passed (at week 0), Noah already had $61 saved. Final answer: The number 16 represents the amount Noah saves per week ($16 per week). The number 61 represents the initial amount Noah had already saved ($61).

5. A local gym charges a $25 membership fee plus $8 per fitness class. The total cost C in dollars for taking x classes is given by the equation C = 8x + 25. If Noah's total bill for the month was $89, how many fitness classes did he attend? Answer: 8 Solution: C = 8x + 25 where C is the total cost in dollars and x is the number of fitness classes. We know Noah's total bill for the month was $89, so C = 89. Substitute C = 89 into the equation.
   Full step-by-step solution

   We are given the equation for the total cost: C = 8x + 25 where C is the total cost in dollars and x is the number of fitness classes. We know Noah's total bill for the month was $89, so C = 89. Step 1: Substitute C = 89 into the equation. 89 = 8x + 25 Step 2: Subtract 25 from both sides to isolate the term with x. 89 - 25 = 8x 64 = 8x Step 3: Divide both sides by 8 to solve for x. 64 / 8 = x 8 = x Step 4: Interpret the result. x = 8 means Noah attended 8 fitness classes. Final answer: 8

6. The equation y = 8x + 25 models the total cost (y) in dollars for a taxi ride based on distance traveled (x) in miles. What does the slope represent in this context? What does the y-intercept represent? Answer: The slope represents the cost per mile ($8 per mile) and the y-intercept represents the initial fee ($25 initial fee) Solution: Identify the slope and y-intercept in the equation y = 8x + 25 - Slope (m) = 8 - Y-intercept (b) = 25 - The slope represents the rate of change of cost with respect to distance - Since x is distance in miles and y is cost in dollars, the slope of 8 means the cost increases by $8 for each…
   Full step-by-step solution

   Step 1: Identify the slope and y-intercept in the equation y = 8x + 25 - Slope (m) = 8 - Y-intercept (b) = 25 Step 2: Interpret the slope in context - The slope represents the rate of change of cost with respect to distance - Since x is distance in miles and y is cost in dollars, the slope of 8 means the cost increases by $8 for each additional mile traveled - Therefore, the slope represents: $8 per mile (cost per mile) Step 3: Interpret the y-intercept in context - The y-intercept is the value of y when x = 0 - When x = 0 (no distance traveled), y = 25 - This represents the initial cost before any miles are traveled - Therefore, the y-intercept represents: $25 initial fee (base fare) The slope represents the cost per mile ($8 per mile) and the y-intercept represents the initial fee ($25 initial fee).

7. A local library is tracking the relationship between the number of hours students spend studying and their test scores. The linear equation y = 2.5x + 65 models this relationship, where x represents study hours and y represents the test score. If Noah studied for 6 hours, what test score would the model predict he would receive? Answer: 80 Solution: y = 2.5x + 65 where x = study hours and y = test score. Identify the value of x from the problem. Noah studied for 6 hours, so x = 6.
   Full step-by-step solution

   We are given the linear equation: y = 2.5x + 65 where x = study hours and y = test score. Step 1: Identify the value of x from the problem. Noah studied for 6 hours, so x = 6. Step 2: Substitute x = 6 into the equation. y = 2.5 * 6 + 65 Step 3: Perform the multiplication first (order of operations). 2.5 * 6 = 15 Step 4: Add the result to 65. y = 15 + 65 Step 5: Calculate the final value. y = 80 So, the model predicts Noah would receive a test score of 80.