➕ Algebra 1 · Systems of Equations

Solving Systems by Graphing practice questions

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Answer key with explanations

1. Which ordered pair is the solution of the system y = x + 3 and y = −2x + 9?

   Answer: (2, 5). Test each pair in both equations. For (2, 5): 5 = 2 + 3 is true and 5 = −2(2) + 9 is true. The pair (5, 2) has its coordinates swapped. The pair (4, 7) fits only the first equation (7 = 4 + 3, but −2(4) + 9 = 1). The pair (1, 7) fits only the second (7 = −2(1) + 9, but 1 + 3 = 4).

2. Is (5, −3) a solution of the system x + y = 2 and x − y = 6?

   Answer: No: it makes x + y = 2 true but x − y = 6 false.. Substitute into each equation. First: 5 + (−3) = 2, true. Second: 5 − (−3) = 8, not 6, so it is false. A solution must make both equations true, so (5, −3) is not a solution of the system. Checking only the first equation would wrongly say yes.

3. Which system has no solution?

   Answer: y = 5x + 2 and y = 5x − 6. A system has no solution when the lines are parallel: the same slope and different y-intercepts. In the first system both slopes are 5 and the y-intercepts are 2 and −6. The second and third systems have different slopes, so each has exactly one solution, (0, 2), where both lines cross the y-axis. The fourth system is the same line twice, which has infinitely many solutions.

4. How many solutions does the system y = 4x − 3 and 8x − 2y = 6 have?

   Answer: Infinitely many solutions. Solve the second equation for y. Subtract 8x from both sides: −2y = −8x + 6. Divide every term by −2: y = 4x − 3. Both equations describe the same line, so every point on it is a solution: infinitely many solutions. Two different lines cannot cross at exactly two points.

5. Which ordered pair is the solution of 2x + y = −3 and y = x + 3?

   Answer: (−2, 1). Rewrite the first equation: y = −2x − 3. It has b = −3 and slope −2. The second has b = 3 and slope 1. The lines cross at (−2, 1). Check both: 2(−2) + 1 = −3 is true, and 1 = −2 + 3 is true. The pair (−3, 0) is on the second line (0 = −3 + 3) but not the first (2(−3) + 0 = −6). The pair (1, −2) swaps the coordinates, and (2, −1) flips both signs.

6. Streaming plan A costs 8 dollars per month plus 1 dollar per movie rental. Plan B costs 4 dollars per month plus 3 dollars per movie rental. Let x be the number of rentals and y the monthly cost in dollars. The lines y = x + 8 and y = 3x + 4 cross at which point, and what does it mean?

   Answer: (2, 10): with 2 rentals, both plans cost 10 dollars. Plan A: 2 + 8 = 10. Plan B: 3(2) + 4 = 10. Both are true, so the lines cross at (2, 10): with 2 rentals both plans cost 10 dollars. In a pair (x, y) the first number is the rentals, so (10, 2) has them swapped. The points (0, 8) and (0, 4) are where each line crosses the y-axis (the monthly fees), but each is on only one line.

7. A student graphs y = x + 6 and y = −x + 1 and reads the crossing point as (−2, 4). What does checking show?

   Answer: The pair fits y = x + 6 but not y = −x + 1, so it is not the solution. The lines cross between grid points, at (−5/2, 7/2).. Test (−2, 4). In y = x + 6: −2 + 6 = 4, true. In y = −x + 1: −(−2) + 1 = 3, not 4, so false. The pair is not the solution, but that does not mean there is none: the slopes 1 and −1 differ, so the lines cross exactly once. Setting x + 6 = −x + 1 gives 2x = −5, so x = −5/2 and y = 7/2. Check: −5/2 + 6 = 7/2 and 5/2 + 1 = 7/2.

8. For which value of k does the system y = kx + 1 and y = 4x − 6 have no solution?

   Answer: k = 4. A system has no solution when the lines are parallel: the same slope and different y-intercepts. With k = 4 the slopes are both 4, and the y-intercepts 1 and −6 are different. For any other k the slopes differ, so the lines cross exactly once. (The value −1/4 is the slope of a line that is perpendicular to y = 4x − 6, not parallel to it.)