PSAT Math Flashcards: Equations With Two Variables

What this deck covers

This deck focuses on Equations With Two Variables, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT 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: Identify the $x$-intercept of the line given by $4x+3y=12$.

Answer: $3$. Substitute $y=0$: $4x=12$, so $x=3$.

Flashcard 2: Identify whether $(2,3)$ is a solution to $x+y=5$.

Answer: Yes. Check: $2+3=5$ ✓

Flashcard 3: Identify the $y$-intercept of the line $4x+2y=10$.

Answer: $5$. Set $x=0$: $2y=10$, so $y=5$.

Flashcard 4: What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. Directly uses the given point and slope to write the equation.

Flashcard 5: What is the slope of the line given by $x=4$?

Answer: Undefined. Vertical lines have no defined slope.

Flashcard 6: What is the equation of a line with slope $2$ and $y$-intercept $-3$?

Answer: $y=2x-3$. Use slope-intercept form with given $m$ and $b$.

Flashcard 7: What is the equation of the line through $(0,-2)$ and $(4,6)$?

Answer: $y=2x-2$. Find slope $m=\frac{6-(-2)}{4-0}=2$, then use $y=mx+b$.

Flashcard 8: Identify the slope of $Ax+By=C$ in terms of $A$ and $B$ (assume $B\ne 0$).

Answer: m=-rac{A}{B}. Solve for $y$ to get $y=-\frac{A}{B}x+\frac{C}{B}$.

Flashcard 9: What condition on slopes shows two lines are parallel?

Answer: Equal slopes: $m_1=m_2$. Parallel lines never intersect, so they have identical slopes.

Flashcard 10: Write the equation in slope-intercept form: $2x+y=7$.

Answer: $y=-2x+7$. Isolate $y$ by subtracting $2x$ from both sides.

Flashcard 11: What does it mean if an ordered pair $(x,y)$ is a solution to an equation in two variables?

Answer: Substitution makes the equation true. The $x$ and $y$ values satisfy the equation.

Flashcard 12: Identify whether $(1,-2)$ is a solution to $2x-y=0$.

Answer: No. Check: $2(1)-(-2)=2+2=4\ne 0$.

Flashcard 13: Find the slope of the line through $(1,3)$ and $(5,1)$.

Answer: -rac{1}{2}. Use slope formula: m=rac{1-3}{5-1}=rac{-2}{4}=-rac{1}{2}.

Flashcard 14: Find the missing value $k$ so that $(3,k)$ satisfies $2x+y=11$.

Answer: $5$. Substitute: $2(3)+k=11$, so $6+k=11$, thus $k=5$.

Flashcard 15: What condition on slopes shows two nonvertical lines are parallel?

Answer: $m_1=m_2$. Parallel lines have same steepness, thus equal slopes.

Flashcard 16: What is the slope of the line given by $Ax+By=C$ in terms of $A$ and $B$?

Answer: $m=-\frac{A}{B}$. Derived by solving $Ax+By=C$ for $y$ to get slope-intercept form.

Flashcard 17: Identify the equation of the vertical line through $x=-3$.

Answer: $x=-3$. Vertical lines have constant $x$-value for all $y$.

Flashcard 18: What is the $y$-intercept of $y=-3x+5$, stated as a point?

Answer: $(0,5)$. When $x=0$, $y=5$.

Flashcard 19: Identify the equation of the line with slope $2$ and $y$-intercept $-3$.

Answer: $y=2x-3$. Direct substitution into $y=mx+b$ form.

Flashcard 20: What does it mean for $(x,y)$ to be a solution to an equation in two variables?

Answer: $(x,y)$ makes the equation true when substituted for $x$ and $y$. The point satisfies the equation when its coordinates are plugged in.

Flashcard 21: What is the slope formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: m=rac{y_2-y_1}{x_2-x_1}. Rise over run between two points gives the slope.

Flashcard 22: What is the solution to the system $2x+y=9$ and $x+y=6$?

Answer: $\left(3,3\right)$. Subtract second from first: $x=3$; substitute to get $y=3$.

Flashcard 23: Which option is the standard form of $y=3x-5$ with integer coefficients?

Answer: $3x-y=5$. Move all terms to one side: $3x-y-5=0$ or $3x-y=5$.

Flashcard 24: What is the equation of the line perpendicular to $y=\frac{1}{3}x-4$ passing through $(0,2)$?

Answer: $y=-3x+2$. Perpendicular slope is $-3$, passes through $(0,2)$.

Flashcard 25: What is the slope of the line in standard form $Ax+By=C$ (with $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

Flashcard 26: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ (with $x_2\ne x_1$)?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

Flashcard 27: Identify the slope of the line $3x-6y=12$.

Answer: $\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$; coefficient of $x$ is the slope.

Flashcard 28: What condition on slopes shows two nonvertical lines are perpendicular?

Answer: $m_1m_2=-1$. Perpendicular slopes multiply to $-1$ (negative reciprocals).

Flashcard 29: Identify the slope of $2x+5y=10$.

Answer: $m=-\frac{2}{5}$. Use $m=-\frac{A}{B}$ with $A=2$, $B=5$.

Flashcard 30: Identify the $x$-intercept of the line $5x-3y=15$.

Answer: $3$. Set $y=0$: $5x=15$, so $x=3$.

Flashcard 31: What does it mean for an ordered pair $(x,y)$ to be a solution to $2x+3y=12$?

Answer: $(x,y)$ makes $2x+3y=12$ true when substituted. The values satisfy the equation when plugged in.

Flashcard 32: Identify the $y$-intercept of $2x+3y=12$.

Answer: $(0,4)$. Set $x=0$: $3y=12$, so $y=4$.

Flashcard 33: Identify the $y$-intercept of the line $y=-3x+5$.

Answer: $b=5$. Constant term in slope-intercept form.

Flashcard 34: What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

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

Flashcard 35: What is the $y$-intercept of the line $Ax+By=C$ when $B\ne 0$?

Answer: $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$ to find where line crosses $y$-axis.

Flashcard 36: What is the $x$-intercept of $5x+y=10$?

Answer: $\left(2,0\right)$. Set $y=0$: $5x=10$, so $x=2$.

Flashcard 37: Identify the slope of a horizontal line written as $y=c$.

Answer: $m=0$. Horizontal lines have no change in $y$.

Flashcard 38: What is the condition for two nonvertical lines with slopes $m_1$ and $m_2$ to be perpendicular?

Answer: $m_1m_2=-1$. Perpendicular slopes multiply to $-1$.

Flashcard 39: What is the $x$-intercept of a line found by setting which variable equal to $0$?

Answer: Set $y=0$. The x-intercept occurs where the line crosses the x-axis.

Flashcard 40: What is the equation of the horizontal line through $y=-3$?

Answer: $y=-3$. Horizontal lines have zero slope and constant $y$-value.

Flashcard 41: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

Flashcard 42: Identify the equation of the line through $(0,4)$ and $(2,0)$.

Answer: $y=-2x+4$. Slope $m=\frac{0-4}{2-0}=-2$; use point-slope form.

Flashcard 43: What is the equation of the line through $(0,2)$ and $(4,6)$ in slope-intercept form?

Answer: $y=x+2$. Slope is $\frac{6-2}{4-0}=1$, passes through $(0,2)$.

Flashcard 44: What is the $x$-intercept of $Ax+By=C$ (assume $A\ne 0$)?

Answer: $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$: $Ax=C$, so $x=\frac{C}{A}$.

Flashcard 45: What is the equation of the vertical line through $x=5$?

Answer: $x=5$. Vertical lines have undefined slope and constant $x$-value.

Flashcard 46: What is the slope-intercept form of a linear equation in two variables?

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

Flashcard 47: What is the equation of the line with slope $-4$ passing through $(0,6)$?

Answer: $y=-4x+6$. Point $(0,6)$ is the y-intercept, so $b=6$.

Flashcard 48: Identify whether $(2,1)$ is a solution to $3x+2y=4$.

Answer: No. Check: $3(2)+2(1)=6+2=8≠4$ ✗

Flashcard 49: What is the $x$-intercept of $Ax+By=C$ when $y=0$?

Answer: $x=\frac{C}{A}$. Set $y=0$ and solve for $x$.

Flashcard 50: What is the slope of the line given by $y=6$?

Answer: $0$. Horizontal lines have zero slope.

Flashcard 51: Find the $x$-intercept of the line $2x+5y=10$.

Answer: $(5,0)$. Set $y=0$: $2x=10$, so $x=5$.

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

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

Flashcard 53: What condition on slopes $m_1$ and $m_2$ means two lines are parallel?

Answer: $m_1=m_2$. Parallel lines have equal slopes.

Flashcard 54: What is the equation of a vertical line through $x=k$?

Answer: $x=k$. All points have the same $x$-coordinate.

Flashcard 55: Solve for $y$ in terms of $x$: $3x+2y=8$.

Answer: y=-rac{3}{2}x+4. Isolate $y$: $2y=8-3x$, then divide by 2.

Flashcard 56: Identify the slope of the line through $(1,2)$ and $(5,10)$.

Answer: $2$. Use m=rac{y_2-y_1}{x_2-x_1}=rac{10-2}{5-1}=rac{8}{4}.

Flashcard 57: What is the condition on slopes for two nonvertical lines to be perpendicular?

Answer: Their slopes satisfy $m_1m_2=-1$. Perpendicular lines meet at 90°; slopes are negative reciprocals.

Flashcard 58: State the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel.

Answer: $m_1=m_2$. Parallel lines have equal slopes.

Flashcard 59: Find the slope of the line $2x+5y=10$.

Answer: $m=-\frac{2}{5}$. Rewrite as $y=-\frac{2}{5}x+2$ to identify slope $m=-\frac{2}{5}$.

Flashcard 60: What is the definition of a solution to an equation in two variables such as $2x+3y=7$?

Answer: An ordered pair $(x,y)$ that makes the equation true. When substituted, both sides of the equation must be equal.

Flashcard 61: Identify the slope of the line given by $3x+6y=12$.

Answer: -rac{1}{2}. Rewrite as y=-rac{1}{2}x+2 to find slope m=-rac{1}{2}.

Flashcard 62: What is the standard form of a linear equation in two variables?

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

Flashcard 63: Find the equation in standard form for the line through $(0,4)$ and $(2,0)$.

Answer: $2x+y=4$. Slope $m=\frac{0-4}{2-0}=-2$; use point-slope form.

Flashcard 64: What is the $y$-intercept of $Ax+By=C$ when $x=0$?

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

Flashcard 65: Find the equation of the line with slope $-2$ passing through $(1,3)$ in slope-intercept form.

Answer: $y=-2x+5$. Use $y-3=-2(x-1)$, then simplify to get $y=-2x+5$.

Flashcard 66: What is the equation in standard form for the line $y=-3x+7$?

Answer: $3x+y=7$. Move $x$ term to left side.

Flashcard 67: Identify whether $(2,-1)$ is a solution to $3x+2y=4$.

Answer: Yes. $3(2)+2(-1)=6-2=4$ ✓

Flashcard 68: Identify the $y$-intercept of the line $4x+5y=10$.

Answer: $\left(0,2\right)$. Set $x=0$: $5y=10$, so $y=2$.

Flashcard 69: Determine whether $(2,3)$ is a solution to $x+y=6$.

Answer: No, because $2+3\ne 6$. Check: $2+3=5\ne 6$.

Flashcard 70: What is the slope of the line given by $Ax+By=C$ (with $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Derived by solving $Ax+By=C$ for $y$ and identifying slope.

Flashcard 71: What is the slope of the line in $Ax+By=C$ (assume $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

Flashcard 72: What is the slope of $Ax+By=C$ (assume $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form to find slope.

Flashcard 73: What is the equation of the line with slope $-2$ passing through $(3,1)$?

Answer: $y=-2x+7$. Use $y-1=-2(x-3)$ and simplify.

Flashcard 74: Identify whether $(1,4)$ is a solution to $2x+y=5$.

Answer: No. Check: $2(1)+4=6≠5$, so not a solution.

Flashcard 75: Write the equation of the line with slope $3$ and $y$-intercept $-2$.

Answer: $y=3x-2$. Direct substitution into $y=mx+b$.

Flashcard 76: What does it mean for an ordered pair $(x,y)$ to be a solution of $ax+by=c$?

Answer: $(x,y)$ makes $ax+by=c$ true when substituted. Substituting the values satisfies the equation.

Flashcard 77: What is the equation in slope-intercept form for $2x+3y=12$?

Answer: y=-rac{2}{3}x+4. Solve for $y$: $3y=12-2x$, then divide by $3$.

Flashcard 78: Identify whether $(2,1)$ is a solution to $3x-2y=4$.

Answer: Yes. Check: $3(2)-2(1)=6-2=4$ ✓

Flashcard 79: What is the equation of the line with slope $3$ and $y$-intercept $-5$?

Answer: $y=3x-5$. Use slope-intercept form with $m=3$ and $b=-5$.

Flashcard 80: What is the $y$-intercept of a line found by setting which variable equal to $0$?

Answer: Set $x=0$. The y-intercept occurs where the line crosses the y-axis.

Flashcard 81: Rewrite $4x+2y=8$ in slope-intercept form.

Answer: $y=-2x+4$. Solve for $y$: $2y=-4x+8$, so $y=-2x+4$.

Flashcard 82: What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

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

Flashcard 83: What is the slope of the line in standard form $Ax+By=C$ when $B\ne 0$?

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

Flashcard 84: What is the slope 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: change in $y$ divided by change in $x$.

Flashcard 85: Find $y$ when $x=-2$ for the equation $y=4x-1$.

Answer: $y=-9$. Substitute: $y=4(-2)-1=-8-1=-9$.

Flashcard 86: What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?

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

Flashcard 87: State the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular.

Answer: $m_1m_2=-1$. Perpendicular slopes multiply to $-1$.

Flashcard 88: What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

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

Flashcard 89: Identify the slope of the line $3x-2y=8$.

Answer: $\frac{3}{2}$. Rewrite as $y=\frac{3}{2}x-4$; coefficient of $x$ is the slope.

Flashcard 90: What is the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 91: What is the $x$-intercept of $y=-2x+8$?

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

Flashcard 92: Find the $x$-intercept of the line $3x+6y=12$.

Answer: $\left(4,0\right)$. Set $y=0$: $3x=12$, so $x=4$.

Flashcard 93: Find the $y$-intercept of the line $2x+5y=10$.

Answer: $(0,2)$. Set $x=0$: $5y=10$, so $y=2$.

Flashcard 94: What is the condition for two lines with slopes $m_1$ and $m_2$ to be parallel?

Answer: $m_1=m_2$. Parallel lines have equal slopes.

Flashcard 95: What is the equation of the line parallel to $y=-\frac{1}{2}x+3$ passing through $(2,1)$?

Answer: $y=-\frac{1}{2}x+2$. Same slope $-\frac{1}{2}$, use point-slope form.

Flashcard 96: What is the equation of a vertical line through $x=a$?

Answer: $x=a$. All points have the same $x$-coordinate $a$.

Flashcard 97: What is the $y$-intercept of $Ax+By=C$ (assume $B\ne 0$) written as a point?

Answer: $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$ to get $y=\frac{C}{B}$.

Flashcard 98: What is the definition of the solution set of an equation in two variables?

Answer: All ordered pairs $(x,y)$ that satisfy the equation. The collection of all points that make the equation true.

Flashcard 99: Find the equation of the line with slope $3$ and $y$-intercept $-2$.

Answer: $y=3x-2$. Use $y=mx+b$ with $m=3$ and $b=-2$.

Flashcard 100: Find the $x$-intercept of $3x-2y=12$.

Answer: $(4,0)$. Set $y=0$: $3x=12$, so $x=4$.