Use Linear Models to Solve Problems - 8th Grade Math

1. What is the slope $m$ if $y$ increases by $12$ when $x$ increases by $3$?

   $m = 4$. Slope equals change in $y$ divided by change in $x$: $\frac{12}{3}$.

2. What is the meaning of a negative slope in a bivariate linear model?

   As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.

3. Identify the meaning of slope $1.5$ in $y = 1.5x + 10$ for sunlight $x$ (hr) and height $y$ (cm).

   Each extra $1$ hr of sunlight predicts $1.5$ cm more height. Slope is the coefficient of $x$.

4. What does the slope $m$ represent in a linear model $y=mx+b$ for bivariate measurement data?

   Rate of change in $y$ for each $1$ unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.

5. What is the predicted change in $y$ when $x$ increases by $6$ in the model $y = 1.2x + 5$?

   $\Delta y = 7.2$. Multiply slope by change in $x$: $1.2 \times 6 = 7.2$.

6. What is the intercept $b$ if $y = 3x + b$ passes through $(4,14)$?

   $b = 2$. Substitute: $14 = 3(4) + b$, so $14 = 12 + b$.

7. Identify the slope and intercept in the model $y=3x+7$.

   $m=3,b=7$. Coefficient of $x$ is slope; constant term is intercept.

8. Identify the slope and intercept in the model $y=-2.5x+10$.

   $m=-2.5,b=10$. Coefficient of $x$ is slope; constant term is intercept.

9. Which variable is the response variable in a model written as $y = mx + b$?

   $y$ is the response (dependent) variable. In $y = mx + b$, $y$ depends on $x$.

10. What does a slope of $0$ mean about the relationship between $x$ and $y$ in a linear model?

    Predicted $y$ does not change as $x$ changes. Zero slope means horizontal line.

11. Identify the slope and intercept in the model $y = -0.5x + 8$.

    $m = -0.5$, $b = 8$. Coefficient of $x$ is slope; constant term is intercept.

12. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y - y_1 = m(x - x_1)$. Rearranges from slope formula through a point.

13. If $y=mx+b$ models dollars vs. hours, what are the units of slope $m$?

    Dollars per hour. Slope units are output units divided by input units.

14. Which equation gives the change in predicted $y$ when $x$ increases by $\triangle x$ in $y=mx+b$?

    $\triangle y=m\triangle x$. Change in $y$ equals slope times change in $x$.

15. If a model has slope $m=-0.8$ kg/week, what does the negative sign mean?

    Each $+1$ week is associated with $-0.8$ kg in predicted mass. Negative slope means inverse relationship between variables.

16. A model is $y = 4x + 20$. What is the predicted change in $y$ when $x$ increases by $3$?

    Increase of $12$. Change is slope times change in $x$: $4 \times 3$.

17. Use $y = 0.5x + 8$ to find the predicted $y$ when $x = 6$.

    $y = 11$. Substitute $x=6$: $y = 0.5(6) + 8 = 3 + 8$.

18. A model is $C=0.12t+3.4$. What is the predicted cost $C$ when $t=10$?

    $4.6$. Substitute $t=10$: $C=0.12(10)+3.4=1.2+3.4=4.6$.

19. Find the slope of the line through $ (2,7) $ and $ (6,15) $.

    $m = 2$. Use $m = \frac{15-7}{6-2} = \frac{8}{4}$.

20. What does the slope $m$ represent in a linear model $y = mx + b$ for bivariate measurement data?

    Rate of change in $y$ for each $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.

21. What is the slope in a linear model $y = mx + b$ and what does it represent in context?

    $m$; the change in $y$ for each $1$-unit increase in $x$. Slope measures rate of change between variables.

22. For $y=-4x+9$, what is the predicted change in $y$ when $x$ increases by $2$?

    $-8$. Use $\triangle y=m\triangle x=-4(2)=-8$.

23. Find the slope of the line through $ (1,10) $ and $ (5,2) $.

    $m = -2$. Use $m = \frac{2-10}{5-1} = \frac{-8}{4}$.

24. What are the units of slope if $x$ is in hours and $y$ is in centimeters?

    $\text{cm/hr}$. Slope units are output units per input unit.

25. What does the intercept $b$ represent in a linear model $y=mx+b$ for bivariate measurement data?

    Predicted value of $y$ when $x=0$. The $y$-intercept is where the line crosses the $y$-axis.

26. What is the predicted $y$ when $x = 5$ for the model $y = 2x + 3$?

    $y = 13$. Substitute $x = 5$: $y = 2(5) + 3 = 10 + 3$.

27. Which variable is the explanatory variable in a model written as $y = mx + b$?

    $x$ is the explanatory (independent) variable. In $y = mx + b$, $x$ is the input variable.

28. Find $b$ in $y = 3x + b$ if the line passes through $(2,11)$.

    $b = 5$. Substitute point: $11 = 3(2) + b$, so $b = 11 - 6$.

29. What does an intercept of $10\ \text{cm}$ mean in a model where $x$ is hours of sunlight and $y$ is height?

    At $x = 0$, predicted height is $10\ \text{cm}$. Intercept is the starting value when input is zero.

30. For $y=0.5x+12$, what is the predicted change in $y$ when $x$ increases by $6$?

    $3$. Use $\triangle y=m\triangle x=0.5(6)=3$.

31. What is the meaning of a slope of $0$ in a bivariate linear model?

    Predicted $y$ stays constant as $x$ changes. Zero slope means a horizontal line.

32. What are the units of intercept $b$ if $y$ is measured in dollars?

    Dollars (same units as $y$). Intercept has the same units as the response variable.

33. Find $b$ in $y = -2x + b$ if the line passes through $(4,1)$.

    $b = 9$. Substitute point: $1 = -2(4) + b$, so $b = 1 + 8$.

34. What is the intercept $b$ if slope $m = 2$ and the line passes through $(0,9)$?

    $b = 9$. When $x = 0$, $y = b$, so intercept equals $y$-coordinate.

35. What is the slope $m$ of the line through $(2,5)$ and $(6,13)$?

    $m = 2$. Use $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

36. What does the phrase "predicted value" mean in a linear model context?

    The model's estimate of $y$ for a given $x$. Linear models predict output from input values.

37. What is the formula to predict $y$ from $x$ using a linear model with slope $m$ and intercept $b$?

    $y=mx+b$. Standard form of a linear equation with slope and intercept.

38. What is the intercept $b$ in context if a model is $D=60t+15$ and $t$ is hours?

    Predicted distance is $15$ when $t=0$ hours. Intercept is the value when the input variable equals zero.

39. If a model has slope $m=1.5$ cm/hr, what does that mean in context?

    Each $+1$ hr is associated with $+1.5$ cm in predicted height. Positive slope means positive association between variables.

40. What is the predicted $y$ when $x=6$ for the model $y=-3x+20$?

    $2$. Substitute $x=6$: $y=-3(6)+20=-18+20=2$.

41. What is the predicted $y$ when $x=0$ for the model $y=1.2x-4$?

    $-4$. When $x=0$, $y=1.2(0)-4=0-4=-4$.

42. What is the predicted $x$ when $y = 17$ for the model $y = 2x + 1$?

    $x = 8$. Solve $17 = 2x + 1$: $16 = 2x$, so $x = 8$.

43. A model is $y = -1.2x + 50$. What is the predicted change in $y$ when $x$ increases by $5$?

    Decrease of $6$. Change is slope times change in $x$: $-1.2 \times 5$.

44. What is the slope formula using two data points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula calculates slope between two points.

45. Use $y = -3x + 12$ to find the predicted $y$ when $x = 5$.

    $y = -3$. Substitute $x=5$: $y = -3(5) + 12 = -15 + 12$.

46. What does a negative slope mean about the association between $x$ and $y$ in a linear model?

    As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.

47. What is the predicted $y$ when $x=4$ for the model $y=2x+5$?

    $13$. Substitute $x=4$: $y=2(4)+5=8+5=13$.

48. What does the intercept $b$ represent in a linear model $y = mx + b$ for bivariate measurement data?

    Predicted value of $y$ when $x = 0$. The $y$-intercept is where the line crosses the $y$-axis.

49. What does a slope of $1.5\ \text{cm/hr}$ mean in a model where $x$ is hours and $y$ is plant height?

    Each additional $1$ hour predicts $1.5\ \text{cm}$ more height. Slope is the rate of change in context.

50. A model is $H=1.5S+20$. What is the predicted height $H$ when $S=6$?

    $29$. Substitute $S=6$: $H=1.5(6)+20=9+20=29$.

51. Which value in $y=mx+b$ is the predicted starting amount (baseline) before any increase in $x$?

    $b$. The intercept represents the initial value before changes.

52. What is the $y$-intercept in a linear model $y = mx + b$ and what does it represent in context?

    $b$; the predicted value of $y$ when $x = 0$. Y-intercept is where the line crosses the y-axis.

53. What is $x$ when $y=0$ for the model $y=-5x+20$?

    $4$. Set $0=-5x+20$, so $5x=20$ and $x=4$.

54. Use $y = 2x + 5$ to find the predicted $y$ when $x = 4$.

    $y = 13$. Substitute $x=4$: $y = 2(4) + 5 = 8 + 5$.

55. Which value is the $y$-intercept in the model $y = 0.75x - 6$?

    $-6$. Y-intercept is the constant term in $y = mx + b$.

56. What is $x$ when $y=0$ for the model $y=2x-8$?

    $4$. Set $0=2x-8$, so $2x=8$ and $x=4$.

57. Identify the meaning of $b$ in $y = 1.5x + 10$ if $x$ is hours of sunlight and $y$ is plant height (cm).

    At $x = 0$, predicted height is $10$ cm. Y-intercept occurs when input is zero.

58. What is the linear model equation if slope $m = 4$ and intercept $b = -7$?

    $y = 4x - 7$. Substitute slope and intercept into $y = mx + b$ form.

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