➕ Algebra 1 · Systems of Equations
Solving Systems by Graphing: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
A solution of a system of two linear equations is an ordered pair that makes both equations true.
✓ True. That is the definition. A pair that makes only one of the equations true is not a solution of the system.
A point that is on only one of the two lines is a solution of the system.
✗ False. A point on one line fits only that line’s equation. A solution of the system has to fit both equations, so it has to be on both lines.
If two lines have the same slope and different y-intercepts, the system has no solution.
✓ True. The lines are parallel. They never meet, so no pair makes both equations true.
Two lines with the same slope always give a system with no solution.
✗ False. If they also have the same y-intercept, they are the same line and the system has infinitely many solutions. Compare the y-intercepts too.
The lines y = −x + 6 and 2y = −2x + 12 are the same line, so the system has infinitely many solutions.
✓ True. Divide every term of 2y = −2x + 12 by 2 to get y = −x + 6. Both equations describe one line, so every point on it is a solution.
If you read a crossing point from a graph, you can skip the check, because a graph is a picture of the exact answer.
✗ False. A drawn line has thickness and your reading can be slightly off. The check in both equations is what tells you the pair is really the solution.
The pair (4, 2) is the solution of 3x − y = 10 and x + 2y = 6.
✗ False. In the first equation, 3(4) − 2 = 10 is true. In the second, 4 + 2(2) = 8, not 6. The pair fits only one equation, so it is not a solution of the system.
Two different lines can cross at exactly two points.
✗ False. Exactly one line passes through two different points. If two lines shared two points, they would be the same line.
Two non-vertical lines with slopes 2 and −3 cross at exactly one point.
✓ True. Different slopes mean the lines are not parallel, so they cross exactly once.
If the two lines do not seem to meet on your graph, the system must have no solution.
✗ False. Lines with different slopes cross somewhere, possibly off the edge of your grid. Compare the slopes before you decide.
The solution of a system in x and y can be written as just “x = 4.”
✗ False. A solution is an ordered pair (x, y). It needs a y-value as well.
The x-coordinate of the crossing of y = 3x − 4 and y = −x + 8 solves 3x − 4 = −x + 8, so it is x = 3.
✓ True. At the crossing the two y-values are equal. Add x to both sides: 4x − 4 = 8. Add 4: 4x = 12, so x = 3.