Linear Models

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Answer Key & Explanations

Linear Models · Grade 8 · Worksheet 2

1. Liam is a photographer who specializes in landscape photos. He uses a linear equation to model the total number of photos he has in his portfolio. The equation is y = 7x + 13, where y represents the total number of photos in his portfolio, and x represents the number of months since he started his business. Interpret the meaning of the slope (7) and the y-intercept (13) in the context of this situation. Answer: The slope of 7 means that Liam adds 7 new photos to his portfolio each month. The y-intercept of 13 means that Liam had 13 photos in his portfolio when he started his business (at 0 months). Solution: Identify the equation and its parts. The equation is y = 7x + 13. In the slope-intercept form y = mx + b, m is the slope and b is the y-intercept.
   Full step-by-step solution

   Step 1: Identify the equation and its parts. The equation is y = 7x + 13. In the slope-intercept form y = mx + b, m is the slope and b is the y-intercept. Step 2: Interpret the slope (m = 7). The slope represents the rate of change. Here, y is the total number of photos and x is the number of months. Since the slope is 7, it means that for every 1 month increase in x, the total number of photos (y) increases by 7. Therefore, the slope of 7 means Liam adds 7 new photos to his portfolio each month. Step 3: Interpret the y-intercept (b = 13). The y-intercept is the value of y when x = 0. Here, x = 0 represents the start of his business (0 months). When x = 0, y = 7(0) + 13 = 13. Therefore, the y-intercept of 13 means that Liam had 13 photos in his portfolio when he started his business. The answer is: The slope of 7 means Liam adds 7 new photos each month. The y-intercept of 13 means Liam started with 13 photos.

2. Isabella is saving money to buy a new laptop. She already has $27 in her savings account. She decides to save $7 each week from her allowance. The total amount of money in her account after w weeks can be modeled by the equation T = 7w + 27, where T represents the total amount in dollars. What is the real-world meaning of the number 27 in this equation? Answer: The $27 is the initial amount Isabella already had in her savings account before she started saving her weekly allowance. Solution: Identify the parts of the linear equation T = 7w + 27. This is in slope-intercept form, y = mx + b, where m is the slope (rate of change) and b is the y-intercept (starting value).
   Full step-by-step solution

   Step 1: Identify the parts of the linear equation T = 7w + 27. This is in slope-intercept form, y = mx + b, where m is the slope (rate of change) and b is the y-intercept (starting value). Step 2: The number 27 is the constant term, which is the y-intercept (b). Step 3: The y-intercept represents the value of T when w = 0. Substitute w = 0 into the equation: T = 7(0) + 27 = 27. Step 4: Interpret this in context. w represents weeks of saving, so w = 0 means before she starts saving her weekly allowance. At that time, the total amount T is $27. Step 5: Therefore, the 27 represents the initial amount Isabella already had in her savings account before she began saving her weekly $7 allowance. Final Answer: The $27 is the initial amount Isabella already had in her savings account before she started saving her weekly allowance.

3. 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.

4. The equation y = 4.25x + 18.50 models the total cost (y) in dollars for a taxi ride, where x is the distance traveled in miles. What does the slope represent? What does the y-intercept represent? Answer: The slope represents the cost per mile ($4.25 per mile) and the y-intercept represents the initial pickup fee ($18.50) Solution: Identify the slope and y-intercept from the equation y = 4.25x + 18.50 Slope (m) = 4.25 Y-intercept (b) = 18.50 Since the equation models taxi ride cost and x represents distance traveled in miles, the slope represents the rate of change of cost with respect to distance.
   Full step-by-step solution

   Step 1: Identify the slope and y-intercept from the equation y = 4.25x + 18.50 Slope (m) = 4.25 Y-intercept (b) = 18.50 Step 2: Interpret the slope in context Since the equation models taxi ride cost and x represents distance traveled in miles, the slope represents the rate of change of cost with respect to distance. This means the taxi charges $4.25 for each additional mile traveled. Step 3: Interpret the y-intercept in context The y-intercept represents the value of y when x = 0 (when no miles are traveled). This means there is an initial pickup fee of $18.50 before any miles are traveled. Step 4: Final interpretation The slope of 4.25 represents the cost per mile ($4.25 per mile). The y-intercept of 18.50 represents the initial pickup fee ($18.50).

5. 3x² - 5x + 7 = ? when x = 4 Answer: 35 Solution: Substitute x = 4 into the expression: 3(4)² - 5(4) + 7 Calculate the exponent first: 4² = 16 Multiply: 3 × 16 = 48 Multiply: 5 × 4 = 20 Rewrite the expression: 48 - 20 + 7 Subtract: 48 - 20 = 28 Add: 28 + 7 = 35 The answer is 35.
   Full step-by-step solution

   Step 1: Substitute x = 4 into the expression: 3(4)² - 5(4) + 7 Step 2: Calculate the exponent first: 4² = 16 Step 3: Multiply: 3 × 16 = 48 Step 4: Multiply: 5 × 4 = 20 Step 5: Rewrite the expression: 48 - 20 + 7 Step 6: Subtract: 48 - 20 = 28 Step 7: Add: 28 + 7 = 35 The answer is 35.

6. Matiu is a plumber who charges for his services. The total cost C in dollars for a job that takes t hours is modeled by the equation C = 48t + 60. What does the number 48 represent in this context, and what does the number 60 represent? Answer: 48 represents the hourly charge in dollars per hour; 60 represents the fixed call-out fee in dollars. Solution: The equation is C = 48t + 60. This is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Step 2: In context, the variable t (time in hours) is the independent variable.
   Full step-by-step solution

   Step 1: The equation is C = 48t + 60. This is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Step 2: In context, the variable t (time in hours) is the independent variable. The coefficient of t, which is 48, tells us how much the total cost increases for each additional hour of work. So, 48 represents Matiu's hourly charge: $48 per hour. Step 3: The constant term 60 is the value of C when t = 0. This represents a fixed cost that is charged even if the job takes zero hours (i.e., just for showing up). So, 60 represents the call-out fee of $60. Final answer: 48 is the hourly rate ($48/hour), and 60 is the fixed call-out fee ($60).

7. A city's public transit system uses the equation R = 1.75p + 250 to model their daily revenue, where R is the total revenue in dollars and p is the number of passenger trips. The transit authority needs to earn at least $1,000 in revenue each day to cover operating costs. What is the minimum number of passenger trips needed to meet this revenue target? Answer: 429 Solution: Step 1: Set up the inequality using the given equation and revenue requirement: 1.75p + 250 ≥ 1000 Step 2: Subtract 250 from both sides: 1.75p ≥ 750 Step 3: Divide both sides by 1.75: p ≥ 750 ÷ 1.75 Step 4: Calculate 750 ÷ 1.75 = 428.57 Step 5: Since we need a whole number of passenger trips and…
   Full step-by-step solution

   Step 1: Set up the inequality using the given equation and revenue requirement: 1.75p + 250 ≥ 1000 Step 2: Subtract 250 from both sides: 1.75p ≥ 750 Step 3: Divide both sides by 1.75: p ≥ 750 ÷ 1.75 Step 4: Calculate 750 ÷ 1.75 = 428.57 Step 5: Since we need a whole number of passenger trips and must meet or exceed $1,000, we round up to the next whole number: 429 Step 6: Verify: 1.75 × 429 + 250 = 750.75 + 250 = 1000.75, which meets the $1,000 requirement The minimum number of passenger trips needed is 429.