25 hard SAT Systems of two linear equations questions

Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.

Math · Algebra~3 per testHard tier

New to this skill? Read the Systems of two linear equations strategy guide first: what it tests, how it is scored, and the traps below in plain English.

What makes these hard

  • Solves fully when adding the equations gives the answer instantly.
  • “No solution” treated as “solve it” instead of matching coefficients.
  • Mixes up which variable the question wants.
Question 1Hard

A sandwich shop sells turkey sandwiches and veggie sandwiches. Pablo bought 3 turkey sandwiches and 7 veggie sandwiches for $52.00, and Rachel bought 5 turkey sandwiches and 2 veggie sandwiches for $44.00. How many turkey sandwiches are in a purchase of 6 turkey sandwiches and 9 veggie sandwiches?

Show the answer and explanation

Why B is right

Let t be the price of a turkey sandwich and v be the price of a veggie sandwich. From Pablo's purchase: 3t+7v=52.003t + 7v = 52.00. From Rachel's purchase: 5t+2v=44.005t + 2v = 44.00. The question asks how many turkey sandwiches are in a purchase of 6 turkey sandwiches and 9 veggie sandwiches, which is simply 6 turkey sandwiches.

Why the others are wrong

  • AThis is the number of veggie sandwiches in the purchase, solving for the wrong quantity.
  • CThis is the total number of items (6 + 9 = 15) in the purchase, using the wrong operation.
  • DThis results from an arithmetic error in counting the turkey sandwiches in the final purchase.
Question 2Hard

2x−7y=142x - 7y = 14 kx−21y=28kx - 21y = 28 For what value of k will the system of equations have no solution?

Show the answer and explanation

Why D is right

For no solution, the equations must represent parallel lines. The coefficient of y in the second equation is -21, which is 3 times the coefficient in the first equation (-7). Therefore k must equal 3 times 2, which is 6. When k=6k = 6, we have 6x−21y=286x - 21y = 28, but 3 times the first equation gives 6x−21y=426x - 21y = 42, so the lines are parallel but not coincident.

Why the others are wrong

  • AThis incorrectly uses the ratio of the constant terms (28/14 = 2) and doubles it, confusing constant ratios with coefficient requirements.
  • BThis applies an incorrect negative sign to a value close to the correct answer, misunderstanding that both equations have the same sign pattern.
  • CThis uses only the multiplication factor relating the y-coefficients without applying it to the x-coefficient of the first equation.
Question 3Hard
$$\begin{cases} y=3x+ry = 3x + r \\ y=−2x+4ry = -2x + 4r \end{cases}$$

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?

Show the answer and explanation

Why C is right

Setting the equations equal: 3x+r=−2x+4r3x + r = -2x + 4r, so 5x=3r5x = 3r and x=3r/5x = 3r/5. Substituting into the first equation: y=3(3r/5)+r=9r/5+5r/5=14r/5y = 3 (3r/5) + r = 9r/5 + 5r/5 = 14r/5. The point is (3r/5(3r/5, 14r/5)14r/5).

Why the others are wrong

  • AThis results from substituting x = r directly without solving for x from the equality of the two expressions.
  • BThis is the result of an arithmetic error when combining terms after substitution.
  • DThis comes from incorrectly subtracting equations instead of setting them equal, leading to an incorrect x-coordinate.
Question 4Hard

2x−6y=182x - 6y = 18 3x+ky=273x + ky = 27 For what value of k will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why A is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 3/2 times the first exactly when k=−9k = -9; the constants agree as well (3/2 × 18 = 27), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Bk = 9 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Ck = -6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 3 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 5Hard

2x+7y=282x + 7y = 28 kx+21y=42kx + 21y = 42 For what value of k will the system of equations above have no solution?

Show the answer and explanation

Why B is right

For no solution, the lines must be parallel but not coincident. The coefficient of y in the second equation is 3 times that in the first (21 = 3 × 7). For parallel lines, k must also be 3 times the coefficient of x in the first equation: k=3×2=6k = 3 \times 2 = 6. The constant terms 42 and 84 (which would be 3 × 28) are not proportional in the same way, so k=6k = 6 gives no solution.

Why the others are wrong

  • AThis results from using the scaling factor directly without applying it to the coefficient of x.
  • CThis results from incorrectly using the coefficient of y from the first equation.
  • DThis results from a sign error when computing the proportional coefficient.
Question 6Hard
$$\begin{cases} y=−5x+11ry = -5x + 11r \\ y=3x−5ry = 3x - 5r \end{cases}$$

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?

Show the answer and explanation

Why B is right

Setting the equations equal: −5x+11r=3x−5r-5x + 11r = 3x - 5r, so 16r=8x16r = 8x and x=2rx = 2r. Substituting into the second equation: y=3(2r)−5r=6r−5r=ry = 3 (2r) - 5r = 6r - 5r = r. The point is (2r, r).

Why the others are wrong

  • AThis results from incorrectly assuming x = 3r by confusing the coefficient in the second equation.
  • CThis comes from a sign error when computing y, incorrectly calculating 6r - 5r as -r.
  • DThis is the result of incorrectly setting x = 16r and computing an incorrect y-value from that.
Question 7Hard

9x+6y=369x + 6y = 36 3x+ky=183x + ky = 18 In the given system of equations, k is a constant. For what value of k does the system have no solution?

Show the answer and explanation

Why B is right

For no solution the lines must be parallel but distinct. The x- and y-coefficients become proportional (the second equation's coefficients are 1/3 times the first's) when k=2k = 2, but the constant does not follow that ratio: 1/3 × 36 = 12, not 18. Parallel lines with different intercepts never intersect. Any other value of k gives a single unique solution.

Why the others are wrong

  • Ak = 1 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Ck = 3 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Dk = 6 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
Question 8Hard

6x+8y=426x + 8y = 42 3x+ky=183x + ky = 18 For what value of k will the system of equations above have no solution?

Show the answer and explanation

Why D is right

For no solution the lines must be parallel but distinct. The x- and y-coefficients become proportional (the second equation's coefficients are 1/2 times the first's) when k=4k = 4, but the constant does not follow that ratio: 1/2 × 42 = 21, not 18. Parallel lines with different intercepts never intersect. Any other value of k gives a single unique solution.

Why the others are wrong

  • Ak = 8 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Bk = 2 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Ck = -4 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
Question 9Hard
$$\begin{cases} y=7x+ry = 7x + r \\ y=3x+9ry = 3x + 9r \end{cases}$$

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?

Show the answer and explanation

Why D is right

Setting the equations equal: 7x+r=3x+9r7x + r = 3x + 9r, so 4x=8r4x = 8r and x=2rx = 2r. Substituting into the first equation: y=7(2r)+r=14r+r=15ry = 7 (2r) + r = 14r + r = 15r. The point is (2r, 15r).

Why the others are wrong

  • AThis results from incorrectly adding equations instead of setting them equal to find the intersection.
  • BThis comes from incorrectly assuming x = r without solving the system properly.
  • CThis is the result of an arithmetic error when computing y, incorrectly calculating 14r + r as 14r.
Question 10Hard

4x−3y=124x - 3y = 12 −8x+ky=−24-8x + ky = -24 For what value of k will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why C is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is -2 times the first exactly when k=6k = 6; the constants agree as well (-2 × 12 = -24), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = -6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bk = 3 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 8 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 11Hard

6x+4y=486x + 4y = 48 9x+py=729x + py = 72 For what value of p will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why D is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 3/2 times the first exactly when p=6p = 6; the constants agree as well (3/2 × 48 = 72), so the equations are identical. For any other value of p the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ap = 4 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bp = -6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Cp = 8 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 12Hard
$$\begin{cases} y=2x+3ry = 2x + 3r \\ y=−x+9ry = -x + 9r \end{cases}$$

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?

Show the answer and explanation

Why A is right

Setting the equations equal: 2x+3r=−x+9r2x + 3r = -x + 9r, so 3x=6r3x = 6r and x=2rx = 2r. Substituting into the first equation: y=2(2r)+3r=4r+3r=7ry = 2 (2r) + 3r = 4r + 3r = 7r. The point is (2r, 7r).

Why the others are wrong

  • BThis results from incorrectly setting x = 3r by confusing the coefficient with the solution.
  • CThis comes from incorrectly multiplying the entire result by an extra factor.
  • DThis is the result of an arithmetic error when computing y, incorrectly calculating 4r + 3r as 6r.
Question 13Hard

5x+3y=305x + 3y = 30 kx+9y=75kx + 9y = 75 For what value of k will the system of equations above have no solution?

Show the answer and explanation

Why B is right

For no solution the lines must be parallel but distinct. The x- and y-coefficients become proportional (the second equation's coefficients are 3 times the first's) when k=15k = 15, but the constant does not follow that ratio: 3 × 30 = 90, not 75. Parallel lines with different intercepts never intersect. Any other value of k gives a single unique solution.

Why the others are wrong

  • Ak = 10 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Ck = 3 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Dk = -15 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
Question 14Hard

9x−12y=369x - 12y = 36 kx−8y=24kx - 8y = 24 In the given system of equations, k is a constant. For what value of k does the system have infinitely many solutions?

Show the answer and explanation

Why C is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 2/3 times the first exactly when k=6k = 6; the constants agree as well (2/3 × 36 = 24), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = -6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bk = 3 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 9 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 15Hard
$$\begin{cases} y=6x−ry = 6x - r \\ y=2x+7ry = 2x + 7r \end{cases}$$

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?

Show the answer and explanation

Why D is right

Setting the equations equal: 6x−r=2x+7r6x - r = 2x + 7r, so 4x=8r4x = 8r and x=2rx = 2r. Substituting into the first equation: y=6(2r)−r=12r−r=11ry = 6 (2r) - r = 12r - r = 11r. The point is (2r, 11r).

Why the others are wrong

  • AThis results from incorrectly assuming x = r without solving the system properly.
  • BThis comes from an arithmetic error when computing y, incorrectly calculating 12r - r as 10r.
  • CThis is the result of incorrectly adding equations instead of setting them equal and solving.
Question 16Hard

7x+2y=287x + 2y = 28 kx+6y=72kx + 6y = 72 In the given system of equations, k is a constant. The system has no solution. What is the value of k?

Show the answer and explanation

Why D is right

For no solution the lines must be parallel but distinct. The x- and y-coefficients become proportional (the second equation's coefficients are 3 times the first's) when k=21k = 21, but the constant does not follow that ratio: 3 × 28 = 84, not 72. Parallel lines with different intercepts never intersect. Any other value of k gives a single unique solution.

Why the others are wrong

  • Ak = 7 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Bk = 14 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
  • Ck = 18 leaves the coefficients non-proportional, so the system has one unique solution, not no solution.
Question 17Hard

8x+12y=608x + 12y = 60 6x+ky=456x + ky = 45 In the given system of equations, k is a constant. For what value of k does the system have infinitely many solutions?

Show the answer and explanation

Why D is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 3/4 times the first exactly when k=9k = 9; the constants agree as well (3/4 × 60 = 45), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = 6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bk = 8 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Ck = -9 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 18Hard
$$\begin{cases} y=4x−5ry = 4x - 5r \\ y=−2x+7ry = -2x + 7r \end{cases}$$

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?

Show the answer and explanation

Why C is right

Setting the equations equal: 4x−5r=−2x+7r4x - 5r = -2x + 7r, so 6x=12r6x = 12r and x=2rx = 2r. Substituting into the first equation: y=4(2r)−5r=8r−5r=3ry = 4 (2r) - 5r = 8r - 5r = 3r. The point is (2r, 3r).

Why the others are wrong

  • AThis results from an arithmetic error when computing y, incorrectly calculating 8r - 5r as 2r.
  • BThis comes from incorrectly assuming x = r without solving the system properly.
  • DThis is the result of incorrectly dividing the y-coordinate by 2 during computation.
Question 19Hard

5x−2y=145x - 2y = 14 kx−8y=56kx - 8y = 56 For what value of k will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why C is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 4 times the first exactly when k=20k = 20; the constants agree as well (4 × 14 = 56), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = 10 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bk = 14 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 28 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 20Hard

6x+4y=186x + 4y = 18 ax+6y=27ax + 6y = 27 For what value of a will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why C is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 3/2 times the first exactly when a=9a = 9; the constants agree as well (3/2 × 18 = 27), so the equations are identical. For any other value of a the lines have different slopes and meet at a single point.

Why the others are wrong

  • Aa = 4 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Ba = 6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Da = 12 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 21Hard
$$\begin{cases} y=5x−2ry = 5x - 2r \\ y=x+6ry = x + 6r \end{cases}$$

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?

Show the answer and explanation

Why B is right

Setting the equations equal: 5x−2r=x+6r5x - 2r = x + 6r, so 4x=8r4x = 8r and x=2rx = 2r. Substituting into the second equation: y=2r+6r=8ry = 2r + 6r = 8r. The point is (2r, 8r).

Why the others are wrong

  • AThis results from an arithmetic error when computing the y-coordinate, incorrectly calculating 2r + 6r as 7r.
  • CThis comes from setting x = r without properly solving the system equation.
  • DThis is the result of incorrectly dividing the y-coordinate by 2 when substituting.
Question 22Hard

5x−2y=155x - 2y = 15 20x+cy=6020x + cy = 60 For what value of c will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why A is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 4 times the first exactly when c=−8c = -8; the constants agree as well (4 × 15 = 60), so the equations are identical. For any other value of c the lines have different slopes and meet at a single point.

Why the others are wrong

  • Bc = -2 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Cc = 2 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dc = 8 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 23Hard

9x−4y=639x - 4y = 63 27x+ky=18927x + ky = 189 For what value of k will the system of equations above have infinitely many solutions?

Show the answer and explanation

Why A is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 3 times the first exactly when k=−12k = -12; the constants agree as well (3 × 63 = 189), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Bk = -4 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Ck = 12 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 27 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 24Hard

10x−15y=4510x - 15y = 45 kx−6y=18kx - 6y = 18 In the given system of equations, k is a constant. For what value of k does the system have infinitely many solutions?

Show the answer and explanation

Why C is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 2/5 times the first exactly when k=4k = 4; the constants agree as well (2/5 × 45 = 18), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = -4 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Bk = 2 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
Question 25Hard

3x+6y=183x + 6y = 18 kx+4y=12kx + 4y = 12 For what value of k will the system of equations have infinitely many solutions?

Show the answer and explanation

Why B is right

For infinitely many solutions the two equations must be the same line — one a constant multiple of the other. The second equation is 2/3 times the first exactly when k=2k = 2; the constants agree as well (2/3 × 18 = 12), so the equations are identical. For any other value of k the lines have different slopes and meet at a single point.

Why the others are wrong

  • Ak = 1 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Ck = 3 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.
  • Dk = 6 leaves the coefficients non-proportional, so the system has one unique solution, not infinitely many solutions.

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