Create a practice test about determining if an ordered pair is a solution of a system of equations in two variables.
Understand the Problem
The question is asking for the creation of a practice test focused on determining whether an ordered pair is a solution to a system of equations in two variables. This involves checking if the pair satisfies both equations in the system.
Answer
The ordered pair $(1, 5)$ is not a solution to the system of equations.
Answer for screen readers
The ordered pair $(1, 5)$ is not a solution to the system of equations.
Steps to Solve
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Identify the system of equations Determine the two equations that need to be checked against the ordered pair. For example, let’s consider the system: $$ y = 2x + 3 $$ $$ y = -x + 1 $$
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Substitute the ordered pair into one equation Take the ordered pair, say $(x, y) = (1, 5)$, and substitute it into one of the equations. Let's use the first equation: $$ y = 2x + 3 $$ Substituting $x = 1$ gives: $$ 5 = 2(1) + 3 $$
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Simplify and check Now simplify the right side: $$ 5 = 2 + 3 \implies 5 = 5 $$ This is true, so the ordered pair satisfies the first equation.
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Substitute the ordered pair into the second equation Now we check the second equation with the same ordered pair $(1, 5)$: $$ y = -x + 1 $$ Substituting $x = 1$ gives: $$ 5 = -(1) + 1 $$
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Simplify and check Now simplify the right side: $$ 5 = -1 + 1 \implies 5 = 0 $$ This is false, meaning the ordered pair does not satisfy the second equation.
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Conclusion Since the ordered pair satisfies the first equation but not the second, it is not a solution to the system of equations.
The ordered pair $(1, 5)$ is not a solution to the system of equations.
More Information
When determining if an ordered pair is a solution to a system of equations, it must satisfy all equations in the system. If it does not satisfy even one equation, it is not a solution.
Tips
- Forgetting to substitute into both equations can lead to incorrect conclusions about the ordered pair.
- Miscalculating during the substitution or simplification steps can yield false results.
- Not remembering that both equations must be satisfied for the ordered pair to be considered a solution.
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