ACT Math Flashcards: Linear Equations

What this deck covers

This deck focuses on Linear Equations, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 3: What is the equation of a line with slope $m$ and y-intercept $b$?

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

Flashcard 4: Identify the slope of the line passing through $(0, 0)$ and $(2, 3)$.

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

Flashcard 5: What is the slope of a line parallel to a line with slope $m$?

Answer: $m$. Parallel lines have identical slopes.

Flashcard 6: What is the slope of a vertical line?

Answer: Undefined. Division by zero in slope formula.

Flashcard 7: Find the equation of a line with slope $\frac{1}{3}$ through $(0, 5)$.

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

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

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

Flashcard 9: What condition on slopes $m_1$ and $m_2$ guarantees two nonvertical lines are perpendicular?

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

Flashcard 10: What is the x-intercept of $y=2x-8$?

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

Flashcard 11: Convert $y = \frac{1}{2}x + 3$ to standard form.

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

Flashcard 12: What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ if $x_1=x_2$?

Answer: Undefined. Vertical line has undefined slope.

Flashcard 13: What is the equation in slope-intercept form of $2x-y=5$?

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

Flashcard 14: What is the slope of the line $Ax+By=C$ in terms of $A$ and $B$?

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

Flashcard 15: Solve for $y$: $3x + 4y = 12$.

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

Flashcard 16: Convert $y - 2 = -\frac{1}{2}(x + 4)$ to standard form.

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

Flashcard 17: Identify the slope of the line passing through $(0, 0)$ and $(2, 3)$.

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

Flashcard 18: What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ if $y_1=y_2$?

Answer: $0$. Horizontal line has zero slope.

Flashcard 19: Identify the solution for $\frac{x}{4}+3=5$.

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

Flashcard 20: Find the equation of a line with slope $\frac{1}{3}$ through $(0, 5)$.

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

Flashcard 21: What is the slope of $6x+3y=12$?

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

Flashcard 22: Convert $y = \frac{1}{2}x + 3$ to standard form.

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

Flashcard 23: What is the slope of a line parallel to $y = 2x - 5$?

Answer:

1. Parallel lines have identical slopes.

Flashcard 24: Identify the equation of the line with undefined slope passing through $(4,-2)$.

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

Flashcard 25: State the formula to find the slope from two points $(x_1, y_1)$ and $(x_2, y_2)$.

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

Flashcard 26: Identify the solution for $2x+3=2x-5$.

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

Flashcard 27: Find the slope of a line parallel to $2x + 5y = 10$.

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

Flashcard 28: Find the equation of the line with slope -5 through $(1, -2)$.

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

Flashcard 29: What is the midpoint of the segment with endpoints $(2,1)$ and $(8,9)$?

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

Flashcard 30: Identify the solution for $\frac{x}{4}+3=5$.

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

Flashcard 31: What is the slope formula between points $(x_1, y_1)$ and $(x_2, y_2)$?

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

Flashcard 32: Convert $2x + 3y = 6$ to slope-intercept form.

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

Flashcard 33: What is the slope of a vertical line?

Answer: Undefined. Division by zero in slope formula.

Flashcard 34: What is the equation of a line perpendicular to $y = 3x - 1$ through $(2, 5)$?

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

Flashcard 35: What is the slope of $6x+3y=12$?

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

Flashcard 36: Convert $2x + 3y = 6$ to slope-intercept form.

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

Flashcard 37: What is the slope of the line through $(2,5)$ and $(6,1)$?

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

Flashcard 38: What is the x-intercept of $y=2x-8$?

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

Flashcard 39: What is the equation of a horizontal line through $(3, -2)$?

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

Flashcard 40: What is the y-intercept of a line given in standard form $Ax+By=C$?

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

Flashcard 41: What is the slope of a horizontal line?

Answer: $0$. No vertical change means zero slope.

Flashcard 42: Identify the slope in the equation $y = 3x + 7$.

Answer:

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

Flashcard 43: What is the x-intercept of the equation $7x - 3y = 21$?

Answer:

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

Flashcard 44: What is the equation of the line perpendicular to $y=\frac{1}{3}x+2$ through $(0,0)$?

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

Flashcard 45: Determine the y-intercept of $y = -4x + 7$.

Answer:

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

Flashcard 46: Determine the slope of the line passing through $(1,2)$ and $(4,8)$.

Answer:

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

Flashcard 47: What is the y-intercept in the equation $y = -4x + 9$?

Answer:

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

Flashcard 48: Identify the slope of the line $x = 4$.

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

Flashcard 49: What is the slope of a line parallel to a line with slope $m$?

Answer: $m$. Parallel lines have identical slopes.

Flashcard 50: What is the equation of a horizontal line through $y=b$?

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

Flashcard 51: What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 52: What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ if $x_1=x_2$?

Answer: Undefined. Vertical line has undefined slope.

Flashcard 53: Find the x-intercept of the equation $4x - 2y = 8$.

Answer:

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

Flashcard 54: Convert $y - 3 = 2(x - 1)$ to slope-intercept form.

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

Flashcard 55: What is the y-intercept of $6x+3y=12$?

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

Flashcard 56: What is the distance between $(0,0)$ and $(3,4)$?

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

Flashcard 57: What is the equation of the line with slope 3 and y-intercept -4?

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

Flashcard 58: What is the equation of the line with slope 3 and y-intercept -4?

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

Flashcard 59: Which form is the equation $y = -\frac{5}{2}x + 3$ in?

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

Flashcard 60: What is the equation in slope-intercept form of $2x-y=5$?

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

Flashcard 61: What is the slope of a horizontal line?

Answer: $0$. No vertical change means zero slope.

Flashcard 62: Identify the solution set type for $4(x+1)=4x+4$.

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

Flashcard 63: What is the x-intercept of the equation $7x - 3y = 21$?

Answer:

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

Flashcard 64: Determine the slope of the line passing through $(1,2)$ and $(4,8)$.

Answer:

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

Flashcard 65: What is the slope of the line that passes through $(2, 3)$ and $(2, -1)$?

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

Flashcard 66: What is the equation of the line through $(1,4)$ and $(3,8)$?

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

Flashcard 67: What is the slope of a line perpendicular to a line with slope $m$ (with $m\ne 0$)?

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

Flashcard 68: Convert $y - 2 = -\frac{1}{2}(x + 4)$ to standard form.

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

Flashcard 69: What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 70: What is the slope of a line perpendicular to a line with slope $m$ (with $m\ne 0$)?

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

Flashcard 71: What is the equation of the line through $(1,4)$ and $(3,8)$?

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

Flashcard 72: What is the equation of a vertical line through $(5, 7)$?

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

Flashcard 73: What is the equation of the line perpendicular to $y=\frac{1}{3}x+2$ through $(0,0)$?

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

Flashcard 74: Identify the solution for $3x-7=11$.

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

Flashcard 75: Identify the slope of a line perpendicular to $y = 2x + 5$.

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

Flashcard 76: What is the equation of a line with slope $m$ passing through $(0,b)$?

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

Flashcard 77: Convert $y - 3 = 2(x - 1)$ to slope-intercept form.

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

Flashcard 78: What is the equation of a line perpendicular to $y = 3x - 1$ through $(2, 5)$?

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

Flashcard 79: Identify the solution for $3x-7=11$.

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

Flashcard 80: Identify the slope of a line perpendicular to $y = 2x + 5$.

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

Flashcard 81: What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 82: What condition on slopes $m_1$ and $m_2$ guarantees two nonvertical lines are parallel?

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

Flashcard 83: Find the x-intercept of $y = 7x - 14$.

Answer:

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

Flashcard 84: What is the y-intercept of a line given in standard form $Ax+By=C$?

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

Flashcard 85: What is the standard form of a linear equation (with integer coefficients)?

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

Flashcard 86: Identify the solution for $2x+3=2x-5$.

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

Flashcard 87: Determine the y-intercept of $y = -4x + 7$.

Answer:

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

Flashcard 88: What is the y-intercept in the equation $y = -4x + 9$?

Answer:

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

Flashcard 89: What is the slope of the line that passes through $(2, 3)$ and $(2, -1)$?

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

Flashcard 90: What is the slope of a line parallel to $y = 2x - 5$?

Answer:

1. Parallel lines have identical slopes.

Flashcard 91: What is the slope-intercept form of a linear equation?

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

Flashcard 92: What is the midpoint of the segment with endpoints $(2,1)$ and $(8,9)$?

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

Flashcard 93: What is the equation of the line parallel to $y=-2x+1$ through $(3,0)$?

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

Flashcard 94: What is the slope of the line $Ax+By=C$ in terms of $A$ and $B$?

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

Flashcard 95: What is the equation of a horizontal line through $(3, -2)$?

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

Flashcard 96: Write the equation of a line with zero slope through $(4, -3)$.

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

Flashcard 97: Find the slope of a line parallel to $2x + 5y = 10$.

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

Flashcard 98: Identify the y-intercept of $2x + 3y = 6$.

Answer:

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

Flashcard 99: What is the slope of the line $3x - 4y = 12$?

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

Flashcard 100: What is the standard form of a linear equation (with integer coefficients)?

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