Linear Equations - ACT Math

1. What is the distance between $(0,0)$ and $(3,4)$?

   $5$. Use distance formula: $\sqrt{3^2+4^2}=5$.

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

   $y=mx+b$. Direct application of slope-intercept form.

3. Identify the slope of the line passing through $ (0, 0) $ and $ (2, 3) $.

   $\frac{3}{2}$. Use slope formula: $\frac{3-0}{2-0} = \frac{3}{2}$.

4. Find the equation of a line with slope $\frac{1}{3}$ through $(0, 5)$.

   $y = \frac{1}{3}x + 5$. Line passes through y-intercept $(0,5)$ with given slope.

5. Convert $4x - y = 8$ to slope-intercept form.

   $y = 4x - 8$. Solve for $y$: $y = 4x - 8$.

6. What condition on slopes $m_1$ and $m_2$ guarantees two nonvertical lines are perpendicular?

   $m_1m_2=-1$. Product of slopes equals $-1$ for perpendicular lines.

7. What is the x-intercept of $y=2x-8$?

   $4$. Set $y=0$ and solve: $0=2x-8$.

8. Convert $y = \frac{1}{2}x + 3$ to standard form.

   $x - 2y = -6$. Multiply by 2: $2y = x + 6$, rearrange to $x - 2y = -6$.

9. What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ if $x_1=x_2$?

   Undefined. Vertical line has undefined slope.

10. What is the equation in slope-intercept form of $2x-y=5$?

    $y=2x-5$. Solve for $y$: $y=2x-5$.

11. Convert $y - 2 = -\frac{1}{2}(x + 4)$ to standard form.

    $x + 2y = 0$. Expand and rearrange to $Ax + By = C$ form.

12. Identify the slope of the line passing through $(0, 0)$ and $(2, 3)$.

    $\frac{3}{2}$. Use slope formula: $\frac{3-0}{2-0} = \frac{3}{2}$.

13. Identify the solution for $\frac{x}{4}+3=5$.

    $x=8$. Subtract $3$, then multiply by $4$.

14. What is the slope of $6x+3y=12$?

    $-2$. Convert to slope-intercept: $y=-2x+4$.

15. What is the slope of a line parallel to $y = 2x - 5$?

    2. Parallel lines have identical slopes.

16. Identify the equation of the line with undefined slope passing through $(4,-2)$.

    $x=4$. Vertical line through given x-coordinate.

17. State the formula to find the slope from two points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $.

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

18. What is the midpoint of the segment with endpoints $(2,1)$ and $(8,9)$?

    $(5,5)$. Average coordinates: $\left(\frac{2+8}{2},\frac{1+9}{2}\right)$.

19. What is the slope formula between points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

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

20. What is the slope of a vertical line?

    Undefined. Division by zero in slope formula.

21. What is the equation of a line perpendicular to $y = 3x - 1$ through $(2, 5)$?

    $y = -\frac{1}{3}x + \frac{17}{3}$. Perpendicular slope is $-\frac{1}{3}$; use point-slope form.

22. Convert $2x + 3y = 6$ to slope-intercept form.

    $y = -\frac{2}{3}x + 2$. Solve for $y$: $3y = -2x + 6$, so $y = -\frac{2}{3}x + 2$.

23. What is the slope of the line through $ (2,5) $ and $ (6,1) $?

    $-1$. Use slope formula: $\frac{1-5}{6-2}=-1$.

24. What is the equation of a horizontal line through $(3, -2)$?

    $y = -2$. Horizontal lines have constant y-value regardless of x.

25. What is the y-intercept of a line given in standard form $Ax+By=C$?

    $\frac{C}{B}$. Set $x=0$ and solve for $y$ in $Ax+By=C$.

26. What is the x-intercept of the equation $7x - 3y = 21$?

    3. Set $y = 0$: $7x = 21$, so $x = 3$.

27. Determine the slope of the line passing through $(1,2)$ and $(4,8)$.

    2. Use slope formula: $\frac{8-2}{4-1} = \frac{6}{3} = 2$.

28. What is the y-intercept in the equation $y = -4x + 9$?

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

29. Identify the slope of the line $x = 4$.

    undefined. Vertical lines have undefined slope (division by zero).

30. What is the slope of a line parallel to a line with slope $m$?

    $m$. Parallel lines have identical slopes.

31. What is the midpoint formula for endpoints $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average of x-coordinates and y-coordinates.

32. Find the x-intercept of the equation $4x - 2y = 8$.

    2. Set $y = 0$: $4x = 8$, so $x = 2$.

33. What is the distance between $ (0,0) $ and $ (3,4) $?

    $5$. Use distance formula: $\sqrt{3^2+4^2}=5$.

34. What is the equation of the line with slope 3 and y-intercept -4?

    $y = 3x - 4$. Direct application of slope-intercept form $y = mx + b$.

35. What is the slope of a horizontal line?

    $0$. No vertical change means zero slope.

36. What is the equation of the line through $(1,4)$ and $(3,8)$?

    $y=2x+2$. Find slope $2$, then use point-slope form.

37. What is the slope of a line perpendicular to a line with slope $m$ (with $m\ne 0$)?

    $-\frac{1}{m}$. Negative reciprocal of original slope.

38. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

    $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate plane.

39. What is the equation of the line perpendicular to $y=\frac{1}{3}x+2$ through $(0,0)$?

    $y=-3x$. Negative reciprocal of $\frac{1}{3}$ is $-3$.

40. Convert $y - 3 = 2(x - 1)$ to slope-intercept form.

    $y = 2x + 1$. Distribute and simplify: $y - 3 = 2x - 2$, so $y = 2x + 1$.

41. Identify the solution for $3x-7=11$.

    $x=6$. Add $7$ to both sides, then divide by $3$.

42. Identify the slope of a line perpendicular to $y = 2x + 5$.

    $-\frac{1}{2}$. Perpendicular slope is negative reciprocal: $-\frac{1}{2}$.

43. What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average of x-coordinates and y-coordinates.

44. Find the x-intercept of $y = 7x - 14$.

    2. Set $y = 0$: $0 = 7x - 14$, so $x = 2$.

45. What is the standard form of a linear equation (with integer coefficients)?

    $Ax+By=C$. General form with integer coefficients $A$, $B$, and $C$.

46. Identify the solution for $2x+3=2x-5$.

    No solution. Coefficients of $x$ are equal but constants differ.

47. Determine the y-intercept of $y = -4x + 7$.

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

48. What is the slope of the line that passes through $(2, 3)$ and $(2, -1)$?

    undefined. Same x-coordinate means vertical line with undefined slope.

49. What is the slope-intercept form of a linear equation?

    $y = mx + b$. Form where $m$ is slope and $b$ is y-intercept.

50. What is the midpoint of the segment with endpoints $ (2,1) $ and $ (8,9) $?

    $(5,5)$. Average coordinates: $\left(\frac{2+8}{2},\frac{1+9}{2}\right)$.

51. What is the equation of the line parallel to $y=-2x+1$ through $(3,0)$?

    $y=-2x+6$. Same slope $-2$, different y-intercept.

52. What is the slope of the line $Ax+By=C$ in terms of $A$ and $B$?

    $-\frac{A}{B}$. Rearrange $Ax+By=C$ to $y=mx+b$ form.

53. Find the slope of a line parallel to $2x + 5y = 10$.

    $-\frac{2}{5}$. Convert to slope-intercept: $y = -\frac{2}{5}x + 2$, same slope.

54. Identify the y-intercept of $2x + 3y = 6$.

    2. Set $x = 0$: $3y = 6$, so $y = 2$.

55. What is the equation of a line parallel to $y = -\frac{1}{4}x + 2$ through $(0, 3)$?

    $y = -\frac{1}{4}x + 3$. Same slope $-\frac{1}{4}$, different y-intercept through $(0,3)$.

56. Which form is the equation $y = -\frac{5}{2}x + 3$ in?

    Slope-intercept form. Format matches $y = mx + b$ with explicit slope and y-intercept.

57. What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ if $y_1=y_2$?

    $0$. Horizontal line has zero slope.

58. What condition on slopes $m_1$ and $m_2$ guarantees two nonvertical lines are parallel?

    $m_1=m_2$. Equal slopes define parallel nonvertical lines.

59. What is the slope of the line through $(2,5)$ and $(6,1)$?

    $-1$. Use slope formula: $\frac{1-5}{6-2}=-1$.

60. Find the slope of $2y - 4x = 8$.

    2. Convert to slope-intercept: $y = 2x + 4$, slope is 2.

61. Find the equation of the line with slope -5 through $(1, -2)$.

    $y = -5x + 3$. Use point-slope form: $y - (-2) = -5(x - 1)$.

62. What is the equation of a line with slope $m$ passing through $(0,b)$?

    $y=mx+b$. Point $(0,b)$ gives y-intercept $b$.

63. What is the equation of a vertical line through $(5, 7)$?

    $x = 5$. Vertical lines have constant x-value regardless of y.

64. Identify the solution for $5(x-2)=15$.

    $x=5$. Divide by $5$, then add $2$.

65. What is the equation of a vertical line through $x=a$?

    $x=a$. All points have same x-coordinate $a$.

66. State the formula to find the slope from two points $(x_1, y_1)$ and $(x_2, y_2)$.

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

67. What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

68. Write the equation of a line with zero slope through $(4, -3)$.

    $y = -3$. Zero slope means horizontal line with constant y-value.

69. Identify the slope of the line $y = -7$.

    0. Horizontal lines have zero slope (no rise).

70. What is the standard form of a linear equation?

    $Ax + By = C$. Form where $A$, $B$, and $C$ are constants with $A$ and $B$ not both zero.

71. What is the slope of the line $3x - 4y = 12$?

    $\frac{3}{4}$. Convert to slope-intercept: $y = \frac{3}{4}x - 3$, slope is $\frac{3}{4}$.

72. Find $b$ if the line $y = 5x + b$ passes through $(2, 1)$.

    $b = -9$. Substitute $(2,1)$: $1 = 5(2) + b$, so $b = -9$.

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

    $y-y_1=m(x-x_1)$. Uses known point and slope to define the line.

74. What is the point-slope form of a linear equation?

    $y - y_1 = m(x - x_1)$. Form using known point $(x_1, y_1)$ and slope $m$.

75. What is the y-intercept of $6x+3y=12$?

    $4$. Convert to slope-intercept: $y=-2x+4$.

76. Identify the slope in the equation $y = 3x + 7$.

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

77. Identify the solution set type for $4(x+1)=4x+4$.

    Infinitely many solutions. Left side simplifies to right side exactly.

78. What is the y-intercept of $y=-3x+7$?

    $7$. Constant term in slope-intercept form.

79. What is the equation of the line with slope $3$ through $(2,-1)$?

    $y=3x-7$. Use point-slope form with $(2,-1)$.

80. What is the equation of a horizontal line through $y=b$?

    $y=b$. All points have same y-coordinate $b$.

81. Solve for $y$: $3x + 4y = 12$.

    $y = -\frac{3}{4}x + 3$. Solve for $y$: $4y = -3x + 12$, so $y = -\frac{3}{4}x + 3$.

82. What is the value of $k$ if the line $y=kx+1$ passes through $(2,9)$?

    $4$. Substitute $(2,9)$: $9=k(2)+1$, so $k=4$.

83. Identify the slope of the line $y=7$.

    $0$. Horizontal line has zero slope.

84. What is the slope of a line perpendicular to $y = -\frac{1}{3}x + 4$?

    3. Perpendicular slopes are negative reciprocals: $-\frac{1}{-\frac{1}{3}} = 3$.

Study these flashcards