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Linear Models

Grade 8 · Algebra · Worksheet 2

  1. (3x + 15) ÷ 3 = 8 Answer: ______________
  2. The graph below shows the distance (in meters) a drone travels over time (in seconds). The line passes through the points (0, 27) and (7, 97). The equation of the line is y = 10x + 27. What does the slope of 10 represent in this context? What does the y-intercept of 27 represent? Answer: ______________
  3. The equation y = 6.5x + 15 represents the total cost (y) in dollars for a bicycle rental, where x is the number of hours rented. What does the slope represent? What does the y-intercept represent? Answer: ______________
  4. 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: ______________
  5. A local library is tracking the relationship between the number of hours students spend studying and their test scores. They found that for every additional hour of study time, test scores increase by 3.5 points. The library's data shows that students who don't study at all typically score 42 points on the test. Write a linear equation in slope-intercept form that represents this relationship, where x represents study hours and y represents test score. Answer: ______________
  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 the midpoint of the hypotenuse. What is the equation of this line in slope-intercept form? Answer: ______________
  7. 2³ × (7 - 3) + √36 = ? Answer: ______________
  8. The equation y = 3.75x + 12.50 models the total cost (y) in dollars for printing custom t-shirts, where x is the number of shirts printed. What does the slope represent? What does the y-intercept represent? Answer: ______________
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Answer Key & Explanations

Linear Models · Grade 8 · Worksheet 2

  1. (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

  2. The graph below shows the distance (in meters) a drone travels over time (in seconds). The line passes through the points (0, 27) and (7, 97). The equation of the line is y = 10x + 27. What does the slope of 10 represent in this context? What does the y-intercept of 27 represent? Answer: The slope of 10 means the drone travels 10 meters per second. The y-intercept of 27 means the drone started at a distance of 27 meters from the starting point. Solution: The equation is y = 10x + 27, where y is distance in meters and x is time in seconds. The slope is 10. This is the coefficient of x.
    Full step-by-step solution

    Step 1: The equation is y = 10x + 27, where y is distance in meters and x is time in seconds. Step 2: The slope is 10. This is the coefficient of x. Slope represents the rate of change: how much the distance increases for each 1-second increase in time. So the drone travels 10 meters per second. Step 3: The y-intercept is 27. This is the constant term. It represents the value of y when x = 0. So when time is 0 seconds, the drone is already 27 meters from the starting point (its initial distance). The answer: The slope of 10 means the drone travels 10 meters per second. The y-intercept of 27 means the drone started at a distance of 27 meters from the starting point.

  3. The equation y = 6.5x + 15 represents the total cost (y) in dollars for a bicycle rental, where x is the number of hours rented. What does the slope represent? What does the y-intercept represent? Answer: The slope represents the cost per hour ($6.50 per hour) and the y-intercept represents the initial rental fee ($15) Solution: When interpreting linear equations in real-world contexts, the slope (m) typically represents a rate of change, while the y-intercept (b) represents a starting value when the independent variable is zero. For example, in an equation like y = 4x + 10, the slope of 4 would indicate a rate of $4…
    Full step-by-step solution

    When interpreting linear equations in real-world contexts, the slope (m) typically represents a rate of change, while the y-intercept (b) represents a starting value when the independent variable is zero. For example, in an equation like y = 4x + 10, the slope of 4 would indicate a rate of $4 per unit, and the y-intercept of 10 would represent an initial cost before any units are used.

  4. 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

  5. A local library is tracking the relationship between the number of hours students spend studying and their test scores. They found that for every additional hour of study time, test scores increase by 3.5 points. The library's data shows that students who don't study at all typically score 42 points on the test. Write a linear equation in slope-intercept form that represents this relationship, where x represents study hours and y represents test score. Answer: y = 3.5x + 42 Solution: The problem says: "students who don't study at all typically score 42 points on the test." "Don't study at all" means x = 0. When x = 0, y = 42. So b = 42.
    Full step-by-step solution

    Let's break this down step by step. --- **Step 1: Understand the slope-intercept form of a linear equation** The slope-intercept form is: y = mx + b where: m = slope (rate of change) b = y-intercept (value of y when x = 0) x = independent variable (study hours) y = dependent variable (test score) --- **Step 2: Identify the slope (m)** The problem says: "for every additional hour of study time, test scores increase by 3.5 points." This means the slope m = 3.5. --- **Step 3: Identify the y-intercept (b)** The problem says: "students who don't study at all typically score 42 points on the test." "Don't study at all" means x = 0. When x = 0, y = 42. So b = 42. --- **Step 4: Write the equation** Substitute m = 3.5 and b = 42 into y = mx + b: y = 3.5x + 42 --- **Final Answer:** y = 3.5x + 42

  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 the midpoint of the hypotenuse. What is the equation of this line in slope-intercept form? Answer: y = -4/3x + 12 Solution: This problem combines coordinate geometry with properties of triangles. The midpoint of a line segment is found by averaging the x-coordinates and the y-coordinates of the endpoints.
    Full step-by-step solution

    This problem combines coordinate geometry with properties of triangles. The midpoint of a line segment is found by averaging the x-coordinates and the y-coordinates of the endpoints. The slope of a line is the ratio of the vertical change to the horizontal change between two points. Once the slope and a point on the line are known, the equation can be written using the point-slope formula and then simplified to slope-intercept form.

  7. 2³ × (7 - 3) + √36 = ? Answer: 38 Solution: Calculate inside the parentheses: 7 - 3 = 4 Calculate the exponent: 2³ = 2 × 2 × 2 = 8 Perform the multiplication: 8 × 4 = 32 Calculate the square root: √36 = 6 Add the results: 32 + 6 = 38 The answer is 38.
    Full step-by-step solution

    Step 1: Calculate inside the parentheses: 7 - 3 = 4 Step 2: Calculate the exponent: 2³ = 2 × 2 × 2 = 8 Step 3: Perform the multiplication: 8 × 4 = 32 Step 4: Calculate the square root: √36 = 6 Step 5: Add the results: 32 + 6 = 38 The answer is 38.

  8. The equation y = 3.75x + 12.50 models the total cost (y) in dollars for printing custom t-shirts, where x is the number of shirts printed. What does the slope represent? What does the y-intercept represent? Answer: The slope represents the cost per shirt ($3.75 per shirt) and the y-intercept represents the initial setup fee ($12.50) Solution: Identify the slope and y-intercept in the equation y = 3.75x + 12.50 - Slope (m) = 3.75 - Y-intercept (b) = 12.50 - When x increases by 1 (one additional shirt), y increases by 3.75 - This means the slope represents the cost per shirt: $3.75 per shirt - When x = 0 (no shirts printed), y = 12.50…
    Full step-by-step solution

    Step 1: Identify the slope and y-intercept in the equation y = 3.75x + 12.50 - Slope (m) = 3.75 - Y-intercept (b) = 12.50 Step 2: Interpret the slope - When x increases by 1 (one additional shirt), y increases by 3.75 - This means the slope represents the cost per shirt: $3.75 per shirt Step 3: Interpret the y-intercept - When x = 0 (no shirts printed), y = 12.50 - This means the y-intercept represents an initial setup fee: $12.50 Step 4: Final interpretation - Slope: $3.75 per shirt (variable cost) - Y-intercept: $12.50 setup fee (fixed cost)