Linear Models

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Linear Models · Grade 8 · Worksheet 1

1. Emma is training for a charity bike ride. She tracks her training progress and finds that the distance she can cycle in kilometers, y, after x weeks of training is modeled by the linear equation y = 15x + 20. What does the number 15 represent in this context, and what does the number 20 represent? Answer: 15 represents the increase in distance cycled per week (slope), and 20 represents the initial distance she could cycle at week 0 (y-intercept). Solution: Identify the parts of the linear equation y = 15x + 20. It is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
   Full step-by-step solution

   Step 1: Identify the parts of the linear equation y = 15x + 20. It is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Step 2: Here, x represents the number of weeks of training, and y represents the distance Emma can cycle in kilometers. Step 3: The slope is m = 15. Since y increases by 15 for every increase of 1 in x, the slope means that for each additional week of training, Emma can cycle 15 more kilometers. Step 4: The y-intercept is b = 20. This is the value of y when x = 0, which means that at the start of training (week 0), Emma could already cycle 20 kilometers. Final answer: The number 15 represents the increase in cycling distance per week (15 km per week), and the number 20 represents her initial cycling distance at the start of training (20 km).

2. Mere runs a small catering business. The total cost C in dollars to cater for x people at an event is given by the linear equation C = 12x + 80. Interpret the meaning of the slope (12) and the y-intercept (80) in the context of this catering business. Answer: The slope of 12 means the cost increases by $12 for each additional person catered for. The y-intercept of 80 means the fixed cost (such as kitchen setup, equipment rental, or delivery fee) is $80, even if no one attends. Solution: The equation is C = 12x + 80, where C is total cost in dollars and x is number of people. The slope is 12. In this context, it represents the cost per additional person.
   Full step-by-step solution

   Step 1: The equation is C = 12x + 80, where C is total cost in dollars and x is number of people. Step 2: The slope is 12. In this context, it represents the cost per additional person. For every 1 more person catered for, the total cost increases by $12 (this could cover food, plates, napkins, etc. per person). Step 3: The y-intercept is 80. This is the value of C when x = 0. It represents the fixed costs that are charged even if no people attend (e.g., kitchen rental, delivery fee, or basic service charge). Answer: The slope 12 means each additional person adds $12 to the total cost. The y-intercept 80 means the fixed cost is $80 regardless of the number of people.

3. The equation y = 3.75x + 12.50 models the total cost (y) in dollars for a pizza delivery based on the number of toppings (x). What does the slope represent? What does the y-intercept represent? Answer: The slope represents the cost per topping ($3.75 per topping) and the y-intercept represents the base cost of the pizza ($12.50) Solution: Identify the slope and y-intercept from the equation y = 3.75x + 12.50 - Slope (m) = 3.75 - Y-intercept (b) = 12.50 - The slope represents how much y changes when x increases by 1 - In this context, when the number of toppings increases by 1, the total cost increases by $3.75 - Therefore, the…
   Full step-by-step solution

   Step 1: Identify the slope and y-intercept from the equation y = 3.75x + 12.50 - Slope (m) = 3.75 - Y-intercept (b) = 12.50 Step 2: Interpret the slope - The slope represents how much y changes when x increases by 1 - In this context, when the number of toppings increases by 1, the total cost increases by $3.75 - Therefore, the slope represents the cost per topping: $3.75 per topping Step 3: Interpret the y-intercept - The y-intercept represents the value of y when x = 0 - In this context, when there are 0 toppings (x = 0), the total cost is $12.50 - Therefore, the y-intercept represents the base cost of the pizza: $12.50 Final interpretation: The slope represents the cost per topping ($3.75 per topping) and the y-intercept represents the base cost of the pizza ($12.50)

4. Liam is training for a charity walk and tracks his progress. The distance he walks each week, d kilometers, can be modeled by the equation d = 3w + 7, where w is the number of weeks of training. What does the number 3 represent in this equation? Answer: Liam increases his weekly distance by 3 kilometers each week. Solution: The equation d = 3w + 7 is in slope-intercept form (y = mx + b). Here, d is the total weekly distance, w is the number of weeks, 3 is the slope (m), and 7 is the y-intercept (b).
   Full step-by-step solution

   Step 1: The equation d = 3w + 7 is in slope-intercept form (y = mx + b). Here, d is the total weekly distance, w is the number of weeks, 3 is the slope (m), and 7 is the y-intercept (b). Step 2: The slope (3) represents the rate of change: for each additional week (w increases by 1), the distance d increases by 3 kilometers. Step 3: Therefore, the 3 means Liam increases his weekly walking distance by 3 kilometers every week. The y-intercept (7) would represent the distance he walked in week 0 (before training started).

5. Tane is tracking the amount of water in a rain barrel over several days after a storm. The graph shows the water volume (in liters) versus time (in days). The line on the graph passes through the points (0, 81) and (9, 0). What is the equation of this line in slope-intercept form, and what does the slope represent in the context of the problem? Answer: y = -9x + 81; The slope of -9 means the water level decreases by 9 liters per day. Solution: Identify the two points: (0, 81) and (9, 0). The y-intercept is 81 because when x = 0, y = 81. Step 2: Find the slope using the formula: m = (y2 - y1) / (x2 - x1) = (0 - 81) / (9 - 0) = -81 / 9 = -9.
   Full step-by-step solution

   Step 1: Identify the two points: (0, 81) and (9, 0). The y-intercept is 81 because when x = 0, y = 81. Step 2: Find the slope using the formula: m = (y2 - y1) / (x2 - x1) = (0 - 81) / (9 - 0) = -81 / 9 = -9. Step 3: Write the equation in slope-intercept form: y = -9x + 81. Step 4: Interpret the slope: The slope is -9, meaning each day the water volume decreases by 9 liters. The y-intercept of 81 means the barrel started with 81 liters of water. The answer is y = -9x + 81, and the slope represents a decrease of 9 liters per day.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A line is drawn from the vertex at (6,8) to a point on the hypotenuse that divides the triangle into two regions of equal area. What are the coordinates of this point? Answer: (3,4) Solution: When a line is drawn from a vertex to a point on the opposite side of a triangle, it divides the triangle into two smaller triangles.
   Full step-by-step solution

   When a line is drawn from a vertex to a point on the opposite side of a triangle, it divides the triangle into two smaller triangles. If these two triangles have equal areas, the point on the opposite side must be located such that the distances from the vertices create proportional relationships. This concept applies to triangles in coordinate geometry where we can use the area formula and coordinate geometry principles to find specific points that satisfy area conditions.

7. The equation y = 8.5x + 25 models the total cost (y) in dollars for a taxi ride based on distance traveled (x) in miles. What does the slope represent? What does the y-intercept represent? Answer: The slope represents the cost per mile ($8.50 per mile) and the y-intercept represents the initial fee ($25) Solution: In linear equations of the form y = mx + b, the slope (m) represents the rate of change - how much y changes for each unit increase in x.
   Full step-by-step solution

   In linear equations of the form y = mx + b, the slope (m) represents the rate of change - how much y changes for each unit increase in x. The y-intercept (b) represents the initial value of y when x equals zero, which is often a fixed starting cost or base amount.

8. 2³ × (4 + 3) - 5² = ? Answer: 31 Solution: 2³ × (4 + 3) - 5² Handle the exponent 2³ 2³ means 2 × 2 × 2 = 8 So now we have: 8 × (4 + 3) - 5² Handle the parentheses (4 + 3) 4 + 3 = 7 Now we have: 8 × 7 - 5² Handle the exponent 5² 5² = 25 Now we have: 8 × 7 - 25 8 × 7 = 56 Now we have: 56 - 25 56 - 25 = 31 Final Answer: 31
   Full step-by-step solution

   Let's solve the problem step by step. We have: 2³ × (4 + 3) - 5² --- **Step 1: Handle the exponent 2³** 2³ means 2 × 2 × 2 = 8 So now we have: 8 × (4 + 3) - 5² --- **Step 2: Handle the parentheses (4 + 3)** 4 + 3 = 7 Now we have: 8 × 7 - 5² --- **Step 3: Handle the exponent 5²** 5² = 25 Now we have: 8 × 7 - 25 --- **Step 4: Perform multiplication** 8 × 7 = 56 Now we have: 56 - 25 --- **Step 5: Perform subtraction** 56 - 25 = 31 --- **Final Answer:** 31