Derive Linear Equations Using Slope - 8th Grade Math

1. Find $b$ if a line has slope $3$ and passes through $(2,1)$ in $y=mx+b$.

   $b=-5$. Substitute $(2,1)$ into $y=3x+b$: $1=6+b$, so $b=-5$.

2. What is slope $m$ defined as in terms of rise and run on a coordinate plane?

   $m=\frac{\text{rise}}{\text{run}}$. Vertical change over horizontal change between two points.

3. Find the slope between $(-1,4)$ and $(3,-2)$.

   $m=-\frac{3}{2}$. Using $m=\frac{-2-4}{3-(-1)}=\frac{-6}{4}=-\frac{3}{2}$.

4. What is the slope of a vertical line written as $x=c$?

   Slope is undefined. Vertical lines have infinite steepness with no defined ratio.

5. What does slope $m$ represent on a coordinate plane for a non-vertical line?

   Rate of change: $\frac{\text{rise}}{\text{run}}$. Slope measures vertical change per unit horizontal change.

6. Find the slope of a line that rises $12$ units while running $-3$ units.

   $m=-4$. $m=\frac{12}{-3}=-4$ using rise over run.

7. Identify $m$ and $b$ for the line $y=-2x+7$.

   $m=-2$, $b=7$. Read coefficient of $x$ for slope, constant for y-intercept.

8. Find the slope of the line through $(-1,4)$ and $(3,2)$.

   $m=-\frac{1}{2}$. $m=\frac{2-4}{3-(-1)}=\frac{-2}{4}=-\frac{1}{2}$

9. Find the equation in the form $y=mx+b$ with slope $m=3$ and y-intercept $b=-2$.

   $y=3x-2$. Substitute given slope and y-intercept into slope-intercept form.

10. What point is the $y$-intercept $b$ located at on the coordinate plane?

    $(0,b)$. The y-intercept occurs where the line crosses the y-axis ($x=0$).

11. Identify the geometric fact used to show slope is constant on a line using right triangles.

    Right triangles formed are similar. Similar triangles have proportional sides, ensuring constant slope.

12. Find and correct the error: A student says the slope from $ (1,2) $ to $ (4,8) $ is $\frac{3}{6}$.

    Correct slope: $m=\frac{6}{3}=2$. Student reversed the fraction; should be $\frac{\Delta y}{\Delta x}$.

13. What is the value of $b$ in $y=mx+b$ when the line passes through the origin?

    $b=0$. Lines through the origin have y-intercept of zero.

14. Find the equation of the line through $(2,-1)$ and $(2,5)$ (state the line equation).

    $x=2$. Same x-coordinates create a vertical line.

15. Which condition makes slope undefined: $x_2=x_1$ or $y_2=y_1$?

    Undefined when $x_2=x_1$. Vertical lines have no horizontal change, making division by zero.

16. Find the slope of the line through $(2,3)$ and $(6,11)$.

    $m=2$. $m=\frac{11-3}{6-2}=\frac{8}{4}=2$

17. Write $y=mx+b$ for the line with slope $m=\frac{1}{2}$ passing through $(0,-3)$.

    $y=\frac{1}{2}x-3$. Point $(0,-3)$ means $b=-3$; substitute $m$ and $b$.

18. Find $b$ in $y=mx+b$ if $m=3$ and the line passes through $(2,1)$.

    $b=-5$. Substitute $(2,1)$ into $1=3(2)+b$, solving gives $b=-5$.

19. Which property of similar triangles makes slope constant along a non-vertical line?

    Corresponding side ratios are equal. Similar triangles maintain constant ratios between corresponding sides.

20. Find the $y$-intercept $b$ of the line $y=\frac{1}{4}x-6$.

    $b=-6$. The constant term is the y-intercept in slope-intercept form.

21. Identify the transformation that maps one slope triangle on a line to another: scaling, reflection, or rotation?

    Scaling (dilation). Triangles scale proportionally along a line, maintaining slope.

22. What is the slope of any horizontal line written as $y=c$?

    $m=0$. Horizontal lines have no vertical change, so rise = 0.

23. What is the slope of a horizontal line (no change in $y$)?

    $m=0$. No vertical change means rise = 0, so slope = 0.

24. What is the equation of a line through the origin with slope $m$?

    $y=mx$. When $b=0$, the line passes through $(0,0)$ with slope $m$.

25. What is the slope-intercept form of a non-vertical line with slope $m$ and $y$-intercept $b$?

    $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

26. What is the slope of a line given by $y=mx+b$?

    $m$. In $y=mx+b$, the coefficient of $x$ is the slope.

27. Write the equation in the form $y=mx+b$ with slope $m=-2$ and $y$-intercept $b=5$.

    $y=-2x+5$. Substitute given values into slope-intercept form.

28. Write the equation in the form $y=mx$ for a line through the origin with slope $m=\frac{3}{4}$.

    $y=\frac{3}{4}x$. Substitute slope into $y=mx$ for lines through origin.

29. What is the equation of a line through the origin with slope $m$?

    $y=mx$. No y-intercept term since the line passes through $(0,0)$.

30. What ratio equality comes from similar slope triangles on the same line?

    $\frac{\text{rise}_1}{\text{run}_1}=\frac{\text{rise}_2}{\text{run}_2}$. Corresponding sides of similar triangles are proportional.

31. What similarity statement justifies equal slopes from two right triangles on the same line?

    Ratios match: $\frac{\text{rise}}{\text{run}}$ is equal. Similar triangles have equal ratios of corresponding sides.

32. What similarity fact explains why slope is constant on a non-vertical line?

    Right triangles formed are similar, so $\frac{\text{rise}}{\text{run}}$ is constant. Similar triangles have proportional sides.

33. What is the slope formula between $(x_1,y_1)$ and $(x_2,y_2)$ for a non-vertical line?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run gives the rate of vertical change per horizontal change.

34. What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?

    $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

35. Find $b$ for a line with slope $m=2$ that passes through $(3,1)$ in $y=mx+b$.

    $b=-5$. Substitute $(3,1)$: $1=2(3)+b$, so $b=1-6=-5$

36. What is the slope-intercept form when the line passes through the origin?

    $y=mx$ (because $b=0$). Origin means $(0,0)$, so y-intercept $b=0$.

37. What is the $y$-intercept of the line $y=5x-9$?

    $-9$. The constant term in $y=mx+b$ is the y-intercept.

38. Identify the slope of the line through $(0,0)$ and $(5,-10)$.

    $m=-2$. Using $m=\frac{-10-0}{5-0}=\frac{-10}{5}=-2$.

39. Find the slope between $(2,3)$ and $(6,11)$.

    $m=2$. $m=\frac{11-3}{6-2}=\frac{8}{4}=2$

40. What is the slope formula between $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $ on a non-vertical line?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: vertical change divided by horizontal change.

41. What point on the coordinate plane represents the $y$-intercept $b$ of $y=mx+b$?

    $(0,b)$. The y-intercept occurs where $x=0$.

42. Find the slope between $(-1,4)$ and $(3,0)$.

    $m=-1$. $m=\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$

43. What condition on the points makes the slope formula undefined (a vertical line)?

    $x_2-x_1=0$. When denominators equal zero, division is undefined.

44. Find the equation of the line with slope $m=-4$ passing through $(0,7)$.

    $y=-4x+7$. Point $(0,7)$ is the y-intercept, so $b=7$.

45. Find the slope $m$ of the line $y=-\frac{3}{5}x+8$.

    $m=-\frac{3}{5}$. The coefficient of $x$ is the slope in slope-intercept form.

46. What condition on $x_1$ and $x_2$ makes the slope formula undefined (vertical line)?

    $x_2-x_1=0$ (so $x_1=x_2$). Division by zero makes the slope undefined for vertical lines.

47. What is the $y$-intercept of a line given by $y=mx+b$?

    $b$. In $y=mx+b$, the constant term is the y-intercept.

48. What is the slope of a vertical line (no change in $x$)?

    Undefined. Division by zero (run = 0) makes slope undefined.

49. What ratio of corresponding sides in right triangles on a line equals the slope $m$?

    $\frac{\Delta y}{\Delta x}$. Change in y over change in x gives the slope.

50. What is the formula for slope $m$ between $(x_1,y_1)$ and $(x_2,y_2)$ on a non-vertical line?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Change in $y$ over change in $x$ gives the rate of vertical change per horizontal unit.

51. Find the equation of the line through the origin with slope $-3$.

    $y=-3x$. Use $y=mx$ with $m=-3$ for lines through origin.

52. Find the equation of the line with slope $\frac{1}{2}$ and $y$-intercept $-4$.

    $y=\frac{1}{2}x-4$. Substitute slope and y-intercept into $y=mx+b$.

53. Find the $y$-intercept $b$ of the line $y=-\frac{3}{2}x+4$.

    $b=4$. In $y=mx+b$, the constant term is the $y$-intercept.

54. Find the equation in the form $y=mx$ for a line through the origin and $(5,-10)$.

    $y=-2x$. $m=\frac{-10-0}{5-0}=-2$, so $y=-2x$

55. Choose the correct equation for a line through $(0,0)$ and $(4,6)$: $y=\frac{3}{2}x$ or $y=\frac{2}{3}x$.

    $y=\frac{3}{2}x$. Slope $m=\frac{6-0}{4-0}=\frac{6}{4}=\frac{3}{2}$.

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