20 medium SAT Systems of two linear equations 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 Systems of two linear equations questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

5x+2y=295x + 2y = 29 3x+2y=193x + 2y = 19 The solution to the given system of equations is (x, y). What is the value of x−yx - y?

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Why C is right

Subtracting the second equation from the first yields 2x=102x = 10, so x=5x = 5. Substituting into 3x+2y=193x + 2y = 19 gives 15+2y=1915 + 2y = 19, so y=2y = 2. Therefore, x−y=5−2=3x - y = 5 - 2 = 3.

Why the others are wrong

  • AThis may result from a sign error when computing x - y or from incorrectly solving the system.
  • BThis is the value of y alone, not x - y.
  • DThis is the value of x alone, not x - y.
Question 2Medium

Sophia bought 3 notebooks and 5 pens for $18.45. Diego bought 2 notebooks and 3 pens for $11.70. How many notebooks are in Diego's purchase?

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Why C is right

Let n be the price of one notebook and p be the price of one pen. From the given information: 3n+5p=18.453n + 5p = 18.45 and 2n+3p=11.702n + 3p = 11.70. Solving this system yields n=2.85n = 2.85 and p=1.80p = 1.80. The question asks how many notebooks are in Diego's purchase, which is stated directly in the problem as 2.

Why the others are wrong

  • AThis is the number of notebooks in Sophia's purchase, not Diego's purchase.
  • BThis is the number of pens in Sophia's purchase, not the number of notebooks in Diego's purchase.
  • DThis results from misreading the problem or an arithmetic error in identifying the quantities in Diego's purchase.
Question 3Medium
y=5x+ry = 5x + r y=2x−3y = 2x - 3

For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

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Why D is right

Setting the equations equal: 5x+r=2x−35x + r = 2x - 3, so 3x=−3−r3x = -3 - r and x=(−3−r)/3x = (-3 - r)/3. Substituting into y=2x−3y = 2x - 3 gives y=2(−3−r)/3−3=(−6−2r−9)/3=(−15−2r)/3y = 2 (-3 - r)/3 - 3 = (-6 - 2r - 9)/3 = (-15 - 2r)/3.

Why the others are wrong

  • AThis results from dividing by 2 instead of 3 when solving 3x = -3 - r.
  • BThis results from an arithmetic error in computing the y-coordinate, incorrectly simplifying 2x - 3 as (-9 - 2r)/3 instead of (-15 - 2r)/3.
  • CThis uses the correct x-coordinate but incorrectly uses the numerator of x as the y-coordinate without proper substitution.
Question 4Medium

x+3y=15x + 3y = 15 2x+y=102x + y = 10 The solution to the given system of equations is (x, y). What is the value of 3x?

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Why C is right

Multiplying the first equation by 2 yields 2x+6y=302x + 6y = 30. Subtracting the second equation from this yields 5y=205y = 20, so y=4y = 4. Substituting y=4y = 4 into the equation 2x+y=102x + y = 10 yields 2x+4=102x + 4 = 10, so 2x=62x = 6 and x=3x = 3. Therefore, 3x=93x = 9.

Why the others are wrong

  • AThis results from finding the value of x instead of 3x.
  • BThis results from a calculation error when solving for x.
  • DThis results from confusing 3x with the constant term from the first equation.
Question 5Medium

Marcus bought 4 sandwiches and 6 drinks for $37.80. Elena bought 5 sandwiches and 2 drinks for $32.90. How many drinks are in Elena's purchase?

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Why D is right

Let s be the price of one sandwich and d be the price of one drink. From the given information: 4s+6d=37.804s + 6d = 37.80 and 5s+2d=32.905s + 2d = 32.90. Solving this system yields s=5.95s = 5.95 and d=2.25d = 2.25. The question asks how many drinks are in Elena's purchase, which is stated directly in the problem as 2.

Why the others are wrong

  • AThis is the number of drinks in Marcus's purchase, not Elena's purchase.
  • BThis is the number of sandwiches in Elena's purchase, not the number of drinks.
  • CThis results from an arithmetic error or misreading the quantities stated in the problem.
Question 6Medium
y=−x+ry = -x + r y=3x+12y = 3x + 12

For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

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Why A is right

Setting the equations equal: −x+r=3x+12-x + r = 3x + 12, so 4x=r−124x = r - 12 and x=(r−12)/4x = (r - 12)/4. Substituting into y=3x+12y = 3x + 12 gives y=3(r−12)/4+12=(3r−36+48)/4=(3r+12)/4y = 3 (r - 12)/4 + 12 = (3r - 36 + 48)/4 = (3r + 12)/4.

Why the others are wrong

  • BThis results from a sign error when solving 4x = r - 12, incorrectly obtaining x = (12 - r)/4.
  • CThis results from dividing by 2 instead of 4 when solving for x.
  • DThis uses the correct x-coordinate but incorrectly uses the numerator of x as the y-coordinate without proper substitution.
Question 7Medium

x+4y=18x + 4y = 18 3x+2y=163x + 2y = 16 The solution to the given system of equations is (x, y). What is the value of 2x+3y2x + 3y?

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Why A is right

Multiplying the first equation by 3 yields 3x+12y=543x + 12y = 54. Subtracting the second equation gives 10y=3810y = 38, so y=4y = 4. Substituting into the first equation yields x+16=18x + 16 = 18, so x=2x = 2. Therefore 2x+3y=4+12=172x + 3y = 4 + 12 = 17.

Why the others are wrong

  • BThis is the value of x alone, not 2x + 3y.
  • CThis results from using incorrect coefficients when computing the linear combination.
  • DThis results from a sign error when solving for the variables or computing the combination.
Question 8Medium

At a farmer's market, Jenna bought 3 pounds of apples and 5 pounds of oranges for $18.50. Marcus bought 2 pounds of apples and 3 pounds of oranges for $11.50. How many pounds of fruit are in Marcus's purchase?

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Why A is right

Let a be the price per pound of apples and r be the price per pound of oranges. From Jenna's purchase: 3a+5r=18.503a + 5r = 18.50. From Marcus's purchase: 2a+3r=11.502a + 3r = 11.50. Marcus bought 2 + 3 = 5 pounds of fruit total.

Why the others are wrong

  • BThis is the total pounds in Jenna's purchase (3 + 5 = 8), not Marcus's purchase.
  • CThis results from subtracting the quantities instead of adding them (3 - 2 = 1, 5 - 3 = 2, then incorrectly computing).
  • DThis is the result of an arithmetic error, such as computing 2 + 3 = 4 instead of 5.
Question 9Medium
y=7x+ry = 7x + r y=x−4y = x - 4

For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

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Why A is right

Setting the equations equal: 7x+r=x−47x + r = x - 4, so 6x=−4−r6x = -4 - r and x=(−4−r)/6x = (-4 - r)/6. Substituting into y=x−4y = x - 4 gives y=(−4−r)/6−4=(−4−r−24)/6=(−28−r)/6y = (-4 - r)/6 - 4 = (-4 - r - 24)/6 = (-28 - r)/6.

Why the others are wrong

  • BThis results from a sign error when solving 6x = -4 - r, incorrectly obtaining x = (-4 + r)/6.
  • CThis results from dividing by 3 instead of 6 when solving for x.
  • DThis uses the correct x-coordinate but incorrectly uses the numerator as the y-coordinate without proper substitution.
Question 10Medium

3x+4y=223x + 4y = 22 x+2y=10x + 2y = 10 What is the value of x+yx + y in the system of equations?

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Why A is right

Multiplying the second equation by 2 yields 2x+4y=202x + 4y = 20. Subtracting this from the first equation gives x=2x = 2. Substituting x=2x = 2 into x+2y=10x + 2y = 10 yields 2+2y=102 + 2y = 10, so y=4y = 4. Therefore x+y=2+4=6x + y = 2 + 4 = 6.

Why the others are wrong

  • BThis is the value of y alone, not x + y.
  • CThis may result from incorrectly doubling x + y or adding the coefficients rather than the variable values.
  • DThis may result from a sign error when solving the system.
Question 11Medium

At a stationery store, Chen bought 6 notebooks and 4 pens for $22.80. Sarah bought 3 notebooks and 7 pens for $19.05. How many items are in Sarah's purchase?

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Why B is right

Let n be the price per notebook and p be the price per pen. From Chen's purchase: 6n+4p=22.806n + 4p = 22.80. From Sarah's purchase: 3n+7p=19.053n + 7p = 19.05. Sarah bought 3 + 7 = 10 items total.

Why the others are wrong

  • AThis results from an arithmetic error, such as computing 3 + 7 = 9 instead of 10.
  • CThis is the result of adding the larger quantities from each purchase (6 + 7 = 13) or another incorrect operation.
  • DThis is the absolute difference between the quantities Sarah purchased (7 - 3 = 4), which uses the wrong operation.
Question 12Medium

4x+3y=264x + 3y = 26 2x−y=82x - y = 8 The solution to the given system of equations is (x, y). What is the value of x−yx - y?

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Why B is right

Multiplying the second equation by 3 yields 6x−3y=246x - 3y = 24. Adding this to the first equation yields 10x=5010x = 50, so x=5x = 5. Substituting into 2x−y=82x - y = 8 gives 10−y=810 - y = 8, so y=2y = 2. Therefore x−y=5−2=3x - y = 5 - 2 = 3.

Why the others are wrong

  • AThis is the value of y alone, not the difference x - y.
  • CThis is the value of x alone, not the difference x - y.
  • DThis may result from a sign error when solving for one of the variables.
Question 13Medium

At a fruit stand, Alex bought 8 apples and 2 pears for $13.60. Nina bought 5 apples and 6 pears for $17.20. How many pieces of fruit are in Nina's purchase?

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Why D is right

Let a be the price per apple and p be the price per pear. From Alex's purchase: 8a+2p=13.608a + 2p = 13.60. From Nina's purchase: 5a+6p=17.205a + 6p = 17.20. Nina bought 5 + 6 = 11 pieces of fruit total.

Why the others are wrong

  • AThis is the total number of pieces of fruit in Alex's purchase (8 + 2 = 10), not Nina's purchase.
  • BThis results from an arithmetic error, such as computing 5 + 6 = 12 instead of 11.
  • CThis is the absolute difference between the quantities Nina purchased (6 - 5 = 1), which uses the wrong operation.
Question 14Medium

6x−2y=146x - 2y = 14 3x+y=113x + y = 11 The solution to the given system of equations is (x, y). What is the value of x+yx + y?

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Why C is right

Multiplying the second equation by 2 yields 6x+2y=226x + 2y = 22. Adding this to the first equation gives 12x=3612x = 36, so x=3x = 3. Substituting x=3x = 3 into the second equation yields 9+y=119 + y = 11, so y=2y = 2. Therefore x+y=3+2=5x + y = 3 + 2 = 5.

Why the others are wrong

  • AThis is the value of x alone rather than x + y.
  • BThis results from finding 2x instead of x + y.
  • DThis results from a sign error when solving the system.
Question 15Medium

A bakery sells muffins and cookies. Lisa bought 4 muffins and 6 cookies for $14.20. Tom bought 5 muffins and 2 cookies for $11.30. How many items are in Tom's purchase?

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Why C is right

Let m be the price per muffin and c be the price per cookie. From Lisa's purchase: 4m+6c=14.204m + 6c = 14.20. From Tom's purchase: 5m+2c=11.305m + 2c = 11.30. Tom bought 5 + 2 = 7 items total.

Why the others are wrong

  • AThis is the total number of items in Lisa's purchase (4 + 6 = 10), not Tom's purchase.
  • BThis results from an arithmetic error, such as computing 5 + 2 = 8 instead of 7.
  • DThis is the absolute difference between the quantities purchased (5 - 2 = 3), which uses the wrong operation.
Question 16Medium

6x+5y=496x + 5y = 49 2x−3y=−72x - 3y = -7 The solution to the given system of equations is (x, y). What is the value of x+yx + y?

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Why C is right

Multiplying the second equation by 3 yields 6x−9y=−216x - 9y = -21. Subtracting this from the first equation yields 14y=7014y = 70, so y=5y = 5. Substituting y=5y = 5 into the second equation gives 2x−15=−72x - 15 = -7, so x=4x = 4. Therefore x+y=9x + y = 9.

Why the others are wrong

  • AThis is the value of y alone, not x + y.
  • BThis results from finding 2x + y instead of x + y.
  • DThis results from a sign error when adding x and y.
Question 17Medium

4x+y=184x + y = 18 2x−3y=22x - 3y = 2 The solution to the given system of equations is (x, y). What is the value of x−yx - y?

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Why C is right

From the first equation, y=18−4xy = 18 - 4x. Substituting into the second equation gives 2x−3(18−4x)=22x - 3 (18 - 4x) = 2, which simplifies to 2x−54+12x=22x - 54 + 12x = 2, or 14x=5614x = 56. Thus x=4x = 4 and y=2y = 2. Therefore x−y=4−2=2x - y = 4 - 2 = 2.

Why the others are wrong

  • AThis is the value of x + y instead of x - y.
  • BThis is the value of x alone rather than x - y.
  • DThis results from a sign error, possibly computing y - x instead of x - y.
Question 18Medium

3x+4y=223x + 4y = 22 x+2y=10x + 2y = 10 The solution to the given system of equations is (x, y). What is the value of x+yx + y?

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Why A is right

Multiplying the second equation by 2 yields 2x+4y=202x + 4y = 20. Subtracting this from the first equation gives x=2x = 2. Substituting into the second equation yields 2+2y=102 + 2y = 10, so y=4y = 4. Therefore, x+y=2+4=6x + y = 2 + 4 = 6.

Why the others are wrong

  • BThis is the value of y alone, not x + y.
  • CThis results from adding the coefficients incorrectly or misinterpreting the target expression.
  • DThis results from a sign error when performing elimination or substitution.
Question 19Medium

3x+y=143x + y = 14 x+2y=13x + 2y = 13 The solution to the given system of equations is (x, y). What is the value of 4x?

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Why C is right

Multiplying the second equation by 3 yields 3x+6y=393x + 6y = 39. Subtracting the first equation from this yields 5y=255y = 25, so y=5y = 5. Substituting into x+2y=13x + 2y = 13 yields x+10=13x + 10 = 13, so x=3x = 3. Therefore, 4x=124x = 12.

Why the others are wrong

  • AThis results from finding the value of x instead of 4x.
  • BThis results from finding the value of y instead of 4x.
  • DThis results from a sign error in the elimination process.
Question 20Medium

5x−3y=75x - 3y = 7 x+y=3x + y = 3 The solution to the given system of equations is (x, y). What is the value of x−yx - y?

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Why C is right

From the second equation, y=3−xy = 3 - x. Substituting into the first equation yields 5x−3(3−x)=75x - 3 (3 - x) = 7, which simplifies to 5x−9+3x=75x - 9 + 3x = 7, or 8x=168x = 16. Thus x=2x = 2 and y=1y = 1. Therefore x−y=2−1=1x - y = 2 - 1 = 1.

Why the others are wrong

  • AThis is the value of x + y instead of x - y.
  • BThis is the value of x alone rather than x - y.
  • DThis results from a sign error, possibly computing y - x instead of x - y.

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