If one equation is y = 2x + 3 and the other is y = -x + 4, the system has

If one equation is y = 2x + 3 and the other is y = -x + 4, the system has

• A. No solution
• B. One solution
• C. Infinitely many solutions
• D. The lines are the same line

Answer: (B)

When solving a system of linear equations by thinking of each equation as a line, the number of solutions depends on how those lines relate. If they cross, there’s exactly one solution; if they’re parallel, there’s no solution; if they’re the same line, there are infinitely many solutions. Here the first line has slope 2 and the second has slope -1, so they aren’t parallel and will intersect at one point. To find that point, set the equations equal: 2x + 3 = -x + 4. This gives 3x = 1, so x = 1/3. Plugging back, y = 2x + 3 = 2/3 + 3 = 11/3. Therefore, there is one solution, at (1/3, 11/3).

When solving a system of linear equations by thinking of each equation as a line, the number of solutions depends on how those lines relate. If they cross, there’s exactly one solution; if they’re parallel, there’s no solution; if they’re the same line, there are infinitely many solutions.

Here the first line has slope 2 and the second has slope -1, so they aren’t parallel and will intersect at one point. To find that point, set the equations equal: 2x + 3 = -x + 4. This gives 3x = 1, so x = 1/3. Plugging back, y = 2x + 3 = 2/3 + 3 = 11/3. Therefore, there is one solution, at (1/3, 11/3).