Solve the system: x + y = 1 and x − y = 0. How many solutions are there?

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Study for the 8th Grade FAST Mathematics Pre-Algebra Test. Enhance your skills with interactive flashcards and multiple choice questions, each containing hints and explanations to boost your comprehension and readiness for the exam. Get ready to ace your test!

Multiple Choice

Solve the system: x + y = 1 and x − y = 0. How many solutions are there?

Explanation:
Two lines in the plane either cross at one point, run parallel, or are the same line. Here, adding the two equations together eliminates y, giving 2x = 1, so x = 1/2. Substituting back into x + y = 1 yields y = 1 − 1/2 = 1/2. So there is exactly one solution, the point (1/2, 1/2). This happens because the lines have different slopes (one is y = −x + 1 and the other is y = x), so they intersect at a single point rather than being parallel or identical.

Two lines in the plane either cross at one point, run parallel, or are the same line. Here, adding the two equations together eliminates y, giving 2x = 1, so x = 1/2. Substituting back into x + y = 1 yields y = 1 − 1/2 = 1/2. So there is exactly one solution, the point (1/2, 1/2). This happens because the lines have different slopes (one is y = −x + 1 and the other is y = x), so they intersect at a single point rather than being parallel or identical.

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