Linear Models

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Answer Key & Explanations

Linear Models · Grade 8 · Worksheet 1

1. 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.

2. 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.

3. (3x + 15) ÷ 3 = 8 Answer: 3 Solution: (3x + 15) ÷ 3 = 8 Rewrite the division as a fraction. (3x + 15) / 3 = 8 Simplify the left side by dividing each term in the numerator by 3. (3x)/3 + 15/3 = 8 x + 5 = 8 Isolate x by subtracting 5 from both sides.
   Full step-by-step solution

   Let's solve the equation step by step. We start with: (3x + 15) ÷ 3 = 8 --- **Step 1: Rewrite the division as a fraction.** (3x + 15) / 3 = 8 --- **Step 2: Simplify the left side by dividing each term in the numerator by 3.** (3x)/3 + 15/3 = 8 This simplifies to: x + 5 = 8 --- **Step 3: Isolate x by subtracting 5 from both sides.** x + 5 - 5 = 8 - 5 x = 3 --- **Step 4: Check the solution.** Substitute x = 3 into the original equation: (3*3 + 15) ÷ 3 = (9 + 15) ÷ 3 = 24 ÷ 3 = 8 This matches the right side of the original equation, so the solution is correct. --- **Final answer:** x = 3

4. A scientist is studying the growth of bacteria in a lab culture. The number of bacteria, B, after t hours is modeled by the equation B = 5.0 × 10³ × 2^t. How many hours will it take for the bacteria population to reach 3.2 × 10⁵? Answer: 6 Solution: B = 5.0 × 10³ × 2^t We want B = 3.2 × 10⁵.
   Full step-by-step solution

   We are given the equation: B = 5.0 × 10³ × 2^t We want B = 3.2 × 10⁵. --- **Step 1: Set up the equation** Substitute B into the equation: 3.2 × 10⁵ = 5.0 × 10³ × 2^t --- **Step 2: Divide both sides by 5.0 × 10³** (3.2 × 10⁵) / (5.0 × 10³) = 2^t --- **Step 3: Simplify the left-hand side** First, 3.2 / 5.0 = 0.64 Second, 10⁵ / 10³ = 10² = 100 So 0.64 × 100 = 64 Thus: 64 = 2^t --- **Step 4: Express 64 as a power of 2** 64 = 2^6 So: 2^t = 2^6 --- **Step 5: Equate exponents** Since the bases are the same (and greater than 0), t = 6 --- **Step 6: Conclusion** It will take 6 hours for the bacteria population to reach 3.2 × 10⁵. **Final answer:** 6

5. Emma is a DJ who tracks the number of songs she plays during her sets. She notices that the relationship between the time she has been playing (t in hours) and the total number of songs played (s) can be modeled by the linear equation s = 25t + 15. Interpret the meaning of the slope (25) and the y-intercept (15) in the context of this situation. Answer: The slope of 25 means Emma plays 25 songs per hour. The y-intercept of 15 means she had already played 15 songs before starting her timer (or at t = 0). Solution: Identify the slope and y-intercept in the equation s = 25t + 15. The slope is 25, and the y-intercept is 15. Step 2: Interpret the slope (25).
   Full step-by-step solution

   Step 1: Identify the slope and y-intercept in the equation s = 25t + 15. The slope is 25, and the y-intercept is 15. Step 2: Interpret the slope (25). Since t is time in hours and s is number of songs, the slope represents the rate of change: for each additional hour, the number of songs increases by 25. So, Emma plays 25 songs per hour. Step 3: Interpret the y-intercept (15). When t = 0, s = 25(0) + 15 = 15. This means at the start of her set (time zero), she had already played 15 songs (perhaps from a previous set or pre-recorded). The answer: The slope of 25 means Emma plays 25 songs per hour. The y-intercept of 15 means she had already played 15 songs before starting the timer.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by translating the first triangle 4 units to the left and 3 units down, then reflecting it across the x-axis. What are the coordinates of the final vertex that was originally at (6,8)? Answer: (2,-5) Solution: (6, 8) Translation: 4 units left, 3 units down. - Moving left 4 units: subtract 4 from x-coordinate. x becomes: 6 - 4 = 2 - Moving down 3 units: subtract 3 from y-coordinate.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the original vertex** The vertex we are tracking starts at: (6, 8) --- **Step 2: Translate the triangle** Translation: 4 units left, 3 units down. - Moving left 4 units: subtract 4 from x-coordinate. x becomes: 6 - 4 = 2 - Moving down 3 units: subtract 3 from y-coordinate. y becomes: 8 - 3 = 5 After translation: (2, 5) --- **Step 3: Reflect across the x-axis** Reflection across the x-axis changes the sign of the y-coordinate, while the x-coordinate stays the same. - From (2, 5), after reflection: x remains 2 y becomes -5 So the new coordinates are: (2, -5) --- **Step 4: Conclusion** The final coordinates of the vertex that was originally at (6, 8) are: (2, -5) --- **Final answer:** (2, -5)

7. A local coffee shop tracks its daily profit using the equation P = 2.5c - 80, where P represents the profit in dollars and c represents the number of cups of coffee sold. If the shop made a profit of $120 yesterday, how many cups of coffee did they sell? Answer: 80 Solution: P = 2.5c - 80 where P is profit in dollars and c is the number of cups sold. Substitute the given profit into the equation. We are told the profit P = 120.
   Full step-by-step solution

   We are given the profit equation: P = 2.5c - 80 where P is profit in dollars and c is the number of cups sold. Step 1: Substitute the given profit into the equation. We are told the profit P = 120. So: 120 = 2.5c - 80 Step 2: Add 80 to both sides to isolate the term with c. 120 + 80 = 2.5c - 80 + 80 200 = 2.5c Step 3: Divide both sides by 2.5 to solve for c. c = 200 / 2.5 Step 4: Perform the division. 200 divided by 2.5 is the same as 2000 divided by 25 (multiplying numerator and denominator by 10 to make it easier). 2000 / 25 = 80 Step 5: Conclusion. The number of cups sold is 80. Answer: 80

8. (2x + 5) = 17 Answer: 6 Solution: 2x + 5 = 17 Subtract 5 from both sides to isolate the term with x. Reason: The left side has "2x + 5", so we undo the addition of 5 by subtracting 5 from both sides.
   Full step-by-step solution

   We start with the equation: 2x + 5 = 17 Step 1: Subtract 5 from both sides to isolate the term with x. Reason: The left side has "2x + 5", so we undo the addition of 5 by subtracting 5 from both sides. 2x + 5 - 5 = 17 - 5 This simplifies to: 2x = 12 Step 2: Divide both sides by 2 to solve for x. Reason: 2x means 2 times x, so we undo multiplication by dividing both sides by 2. 2x / 2 = 12 / 2 This simplifies to: x = 6 So the solution is x = 6.