20 medium SAT Linear equations · two variables questions

Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.

Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.

Math · Algebra~3 per testMedium tier

None of these repeat the 25 hard Linear equations · two variables questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

In the xy-plane, line p passes through the origin and has a slope of 34\displaystyle \frac{3}{4}. Line q is perpendicular to line p and passes through the point (6, 2). What is the y-intercept of line q?

Show the answer and explanation

Why D is right

Line p has slope 3/4, so perpendicular line q has slope -4/3. Using point-slope form with (6, 2): y−2=−4/3(x−6)y - 2 = -4/3 (x - 6). Expanding: y=−4/3y = -4/3 x+8+2=−4/3x + 8 + 2 = -4/3 x+10x + 10. The y-intercept is 10.

Why the others are wrong

  • AThis incorrectly uses the x-coordinate of the given point with a negative sign.
  • BThis uses the y-coordinate of the given point instead of computing the y-intercept.
  • CThis is the intermediate value before adding the y-coordinate, representing an incomplete calculation.
Question 2Medium

A gym membership includes a monthly fee plus an additional charge for each personal training session. The total monthly cost M, in dollars, for attending s personal training sessions is given by M=60+12sM = 60 + 12s. Which statement is the best interpretation of 12 in this context?

Show the answer and explanation

Why D is right

In the equation M=60+12sM = 60 + 12s, the coefficient 12 is multiplied by s (the number of personal training sessions). This means for each session, $12 is added to the total monthly cost.

Why the others are wrong

  • AThis incorrectly identifies the coefficient of s as the monthly fee, when 60 is the monthly membership fee.
  • BThis represents an arithmetic error, adding 1 to the correct coefficient.
  • CThis incorrectly multiplies 60 and 12 together instead of recognizing that 12 is the rate per session.
Question 3Medium

In the xy-plane, the graph of the equation 2x+8y=402x + 8y = 40 has a y-intercept of (0, k) and an x-intercept of (h, 0). What is the value of k+hk + h?

Show the answer and explanation

Why B is right

For the y-intercept, set x=0x = 0: 8y=408y = 40, so k=5k = 5. For the x-intercept, set y=0y = 0: 2x=402x = 40, so h=20h = 20. Therefore k+h=5+20=25k + h = 5 + 20 = 25.

Why the others are wrong

  • AThis results from only finding the x-intercept or doubling the y-intercept value.
  • CThis results from swapping the calculated intercept values or using incorrect coefficient relationships.
  • DThis incorrectly uses the constant term from the equation without performing the intercept calculations.
Question 4Medium

In the xy-plane, line k passes through the points (0, 7) and (3, 1). What is the slope of line k?

Show the answer and explanation

Why B is right

Using the slope formula m = (y2−y1)(y_{2} - y_{1})/(x2−x1)(x_{2} - x_{1}) with points (0,7)(0, 7) and (3,1)(3, 1): m=(1−7)/(3−0)=−6/3=−2m = (1 - 7)/(3 - 0) = -6/3 = -2.

Why the others are wrong

  • AThis is the numerator of the slope calculation (change in y) without dividing by the denominator (change in x).
  • CThis results from inverting the correct slope, computing (x₂ - x₁)/(y₂ - y₁) instead.
  • DThis is the correct magnitude but with the wrong sign, from an error in subtraction order.
Question 5Medium
In the xy-plane, line l passes through the points (0, 7) and (10, 2).

Which equation defines line l?

Show the answer and explanation

Why D is right

The slope is (2−7)/(10−0)=−5/10=−1/2(2 - 7)/(10 - 0) = -5/10 = -1/2. The y-intercept is 7 since the line passes through (0,7)(0, 7). Therefore the equation is y=−12x+7\displaystyle y = -\frac{1}{2} x + 7.

Why the others are wrong

  • AThis correctly finds the slope as -1/2 but uses -7 as the y-intercept instead of 7, making a sign error.
  • BThis uses the reciprocal of the slope, calculating rise/run as -10/5 = -2 instead of -5/10 = -1/2.
  • CThis correctly finds the slope as -1/2 but uses the y-coordinate of the second point (2) as the y-intercept instead of 7.
Question 6Medium

In the xy-plane, the graph of the equation 2x+5y=202x + 5y = 20 has an x-intercept of (p, 0) and a y-intercept of (0, q). What is the value of p+qp + q?

Show the answer and explanation

Why C is right

For the x-intercept, set y=0y = 0: 2x=202x = 20, so x=10x = 10 and p=10p = 10. For the y-intercept, set x=0x = 0: 5y=205y = 20, so y=4y = 4 and q=4q = 4. Therefore p+q=10+4=14p + q = 10 + 4 = 14.

Why the others are wrong

  • AThis results from incorrectly dividing 20 by the sum of coefficients 2 + 5 instead of finding each intercept separately.
  • BThis results from finding only the x-intercept or confusing the calculation.
  • DThis results from using the constant term 20 directly without computing the intercepts.
Question 7Medium

In the xy-plane, line p is perpendicular to the line with equation 4x+5y=204x + 5y = 20. What is the slope of line p?

Show the answer and explanation

Why D is right

Perpendicular lines have slopes that are negative reciprocals. Rewriting 4x+5y=204x + 5y = 20 in slope-intercept form: 5y=−4x+205y = -4x + 20, so y=−4/5y = -4/5 x+4x + 4. The slope is -4/5, so the perpendicular slope is 5/4.

Why the others are wrong

  • AThis results from taking the negative reciprocal of the original equation's coefficient ratio without first finding the actual slope.
  • BThis is the slope of the given line, not its perpendicular.
  • CThis results from taking only the reciprocal without applying the negative, or from incorrectly handling the sign.
Question 8Medium

A car rental company charges a daily rate plus an additional fee per mile driven. The total cost T, in dollars, for renting a car for one day and driving m miles is given by T=45+0.25mT = 45 + 0.25m. Which statement is the best interpretation of 0.25 in this context?

Show the answer and explanation

Why A is right

In the equation T=45+0.25mT = 45 + 0.25m, the coefficient 0.25 is multiplied by m (the number of miles driven). This means for each mile driven, $0.25 is added to the total cost.

Why the others are wrong

  • BThis incorrectly adds 45 and 0.25 together instead of recognizing that 0.25 is the rate per mile.
  • CThis incorrectly identifies the coefficient of m as the daily rate, when 45 is the daily rental rate.
  • DThis represents an arithmetic error, using 0.20 instead of the correct coefficient 0.25.
Question 9Medium

In the xy-plane, the graph of the equation 7x−14y=847x - 14y = 84 has an x-intercept of (a, 0) and a y-intercept of (0, b). What is the value of a+ba + b?

Show the answer and explanation

Why B is right

For the x-intercept, set y=0y = 0: 7x=847x = 84, so a=12a = 12. For the y-intercept, set x=0x = 0: −14y=84-14y = 84, so b=−6b = -6. Therefore a+b=12+(−6)=6a + b = 12 + (-6) = 6.

Why the others are wrong

  • AThis results from computing a - b instead of a + b, or from sign errors in finding both intercepts.
  • CThis is the value of a (the x-intercept) alone, without adding b.
  • DThis results from taking the absolute value of b and adding, computing a + |b| = 12 + 6.
Question 10Medium

In the xy-plane, line k passes through the points (-2, 7) and (4, -5). What is the slope of line k?

Show the answer and explanation

Why A is right

Using the slope formula m = (y2−y1)(y_{2} - y_{1})/(x2−x1)(x_{2} - x_{1}) with points (−2,7)(-2, 7) and (4,−5)(4, -5) gives m=(−5−7)/(4−(−2))m = (-5 - 7)/(4 - (-2)) = −12/6=−2-12/6 = -2.

Why the others are wrong

  • BThis inverts the slope calculation, using (x₂ - x₁)/(y₂ - y₁) instead of (y₂ - y₁)/(x₂ - x₁).
  • CThis incorrectly uses a positive sign and inverts the calculation.
  • DThis uses a positive sign instead of recognizing the negative slope.
Question 11Medium

In the xy-plane, the graph of the equation 15x−5y=6015x - 5y = 60 has a y-intercept of (0, c) and an x-intercept of (d, 0). What is the value of c−dc - d?

Show the answer and explanation

Why A is right

For the y-intercept, set x=0x = 0: −5y=60-5y = 60, so y=−12y = -12 and c=−12c = -12. For the x-intercept, set y=0y = 0: 15x=6015x = 60, so x=4x = 4 and d=4d = 4. Therefore c−d=−12−4=−16c - d = -12 - 4 = -16.

Why the others are wrong

  • BThis results from incorrectly computing the difference, such as finding (-12 + 4) or confusing the calculation.
  • CThis results from computing d - c instead of c - d, swapping the order of subtraction.
  • DThis results from a sign error, such as computing |c - d| or -(c - d).
Question 12Medium

Line p is defined by 8y−12x=328y - 12x = 32. Line q is perpendicular to line p in the xy-plane. What is the slope of line q?

Show the answer and explanation

Why B is right

Rewriting 8y−12x=328y - 12x = 32 in slope-intercept form: 8y=12x+328y = 12x + 32, so y=3/2y = 3/2 x+4x + 4. The slope of line p is 3/2. Since line q is perpendicular to line p, its slope is the negative reciprocal: -2/3.

Why the others are wrong

  • AThis incorrectly uses the slope of line p itself rather than finding the negative reciprocal for the perpendicular line.
  • CThis finds the reciprocal but fails to apply the negative sign, resulting in a sign error.
  • DThis incorrectly uses the slope of line p with a sign error instead of finding the negative reciprocal.
Question 13Medium

In the xy-plane, the graph of the equation 15x+3y=4515x + 3y = 45 passes through the point (2, k). What is the value of k?

Show the answer and explanation

Why C is right

Substituting x=2x = 2 into 15x+3y=4515x + 3y = 45 gives 15(2)+3y=4515(2) + 3y = 45, so 30+3y=4530 + 3y = 45. Solving for y: 3y=153y = 15, thus y=5y = 5 and k=5k = 5.

Why the others are wrong

  • AThis results from incorrectly calculating 15 - 30 = -15, then -15/3 = -5, reversing the subtraction order.
  • BThis is the coefficient of y in the original equation, not the y-coordinate of the point.
  • DThis is the value of 3y before dividing by 3, failing to complete the final step.
Question 14Medium

In the xy-plane, line p passes through the points (2, 7) and (6, 19). What is the slope of line p?

Show the answer and explanation

Why D is right

Using the slope formula m = (y2−y1)(y_{2} - y_{1})/(x2−x1)(x_{2} - x_{1}) with points (2,7)(2, 7) and (6,19)(6, 19), we get m=(19−7)/(6−2)=12/4=3m = (19 - 7)/(6 - 2) = 12/4 = 3.

Why the others are wrong

  • AThis results from incorrectly computing the reciprocal of the slope.
  • BThis results from a sign error in the slope calculation.
  • CThis results from using the difference in x-coordinates as the slope without dividing by the difference in y-coordinates.
Question 15Medium

In the xy-plane, the graph of the equation 6x+3y=186x + 3y = 18 has a slope of m. What is the value of m?

Show the answer and explanation

Why A is right

Rewriting 6x+3y=186x + 3y = 18 in slope-intercept form by solving for y: 3y=−6x+183y = -6x + 18, so y=−2x+6y = -2x + 6. The slope m is the coefficient of x, which is -2.

Why the others are wrong

  • BThis results from dividing the coefficient of y by the coefficient of x instead of taking the negative ratio.
  • CThis results from taking the positive ratio of coefficients without accounting for the negative sign or swapping the roles of x and y.
  • DThis results from a sign error when solving for the slope.
Question 16Medium

Line t is defined by 12y+18x=7212y + 18x = 72. Line u is perpendicular to line t in the xy-plane. What is the slope of line u?

Show the answer and explanation

Why C is right

Rewriting 12y+18x=7212y + 18x = 72 in slope-intercept form: 12y=−18x+7212y = -18x + 72, so y=−3/2y = -3/2 x+6x + 6. The slope of line t is -3/2. Since line u is perpendicular to line t, its slope is the negative reciprocal: 2/3.

Why the others are wrong

  • AThis incorrectly uses the slope of line t itself rather than finding the negative reciprocal.
  • BThis finds the reciprocal but fails to change the sign correctly, resulting in a sign error.
  • DThis incorrectly uses the reciprocal with the wrong sign, or confuses the original slope with the perpendicular slope.
Question 17Medium

In the xy-plane, the graph of the equation 2x+3y=242x + 3y = 24 passes through the point (6, k). What is the value of k?

Show the answer and explanation

Why B is right

Substituting x=6x = 6 into the equation 2x+3y=242x + 3y = 24 gives 2(6)+3y=242(6) + 3y = 24, so 12+3y=2412 + 3y = 24. Thus 3y=123y = 12 and k=4k = 4.

Why the others are wrong

  • AThis results from dividing 12 by the coefficient of x (2) instead of by the coefficient of y (3).
  • CThis uses the given x-value as the answer without performing the substitution.
  • DThis results from subtracting 24 - 12 = 12 but then dividing incorrectly or applying a sign error.
Question 18Medium

In the xy-plane, line t passes through the origin and has a slope of 34\displaystyle \frac{3}{4}. Which equation defines line t?

Show the answer and explanation

Why A is right

A line through the origin with slope 3/4 has equation y=34\displaystyle y = \frac{3}{4}x. Multiplying both sides by 4 gives 4y=3x4y = 3x, which rearranges to 3x−4y=03x - 4y = 0.

Why the others are wrong

  • BThis results from a sign error when rearranging the slope-intercept form to standard form.
  • CThis swaps the numerator and denominator of the slope, giving slope 4/3 instead of 3/4.
  • DThis combines swapping x and y coefficients with a sign error.
Question 19Medium

In the xy-plane, the graph of the equation 4x−7y=284x - 7y = 28 has an x-intercept at (a, 0). What is the value of a?

Show the answer and explanation

Why D is right

To find the x-intercept, set y=0y = 0 in the equation 4x−7y=284x - 7y = 28. This gives 4x=284x = 28, so x=7x = 7. The x-intercept is (7, 0), therefore a=7a = 7.

Why the others are wrong

  • AThis results from using the coefficient of y instead of solving for the x-intercept.
  • BThis results from incorrectly applying a negative sign and confusing the coefficient.
  • CThis results from using the coefficient of x rather than solving for x when y = 0.
Question 20Medium

In the xy-plane, the graph of the equation 4x−5y=204x - 5y = 20 has a slope of m and a y-intercept of (0, b). What is the value of m?

Show the answer and explanation

Why C is right

To find the slope, rewrite the equation in slope-intercept form y=mx+by = mx + b. Starting with 4x−5y=204x - 5y = 20, subtract 4x from both sides to get −5y=−4x+20-5y = -4x + 20, then divide by -5 to get y=45x−4\displaystyle y = \frac{4}{5} x - 4. The slope m is 4/54/5.

Why the others are wrong

  • AThis results from incorrectly treating the coefficient of x as negative when converting to slope-intercept form.
  • BThis results from incorrectly swapping the coefficients of x and y when determining the slope.
  • DThis results from using only the coefficient of x without accounting for the coefficient of y.

What to do after medium

Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.

Easy Linear equations · two variables questionsHard Linear equations · two variables questionsStart free →

More on Linear equations · two variables

Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.