To determine if $(4, 4)$ is a solution to the system of equations:

$y = \frac{1}{2}x - 2$ $y = \frac{3}{2}x + 2,$

we substitute $x = 4$ and $y = 4$ into both equations and check if the equations hold true.

Substituting into the first equation:

$y = \frac{1}{2}x - 2$ $4 = \frac{1}{2}(4) - 2$ $4 = 2 - 2$ $4 \neq 0$ This equation is not satisfied, so $(4, 4)$ is not a solution to the system of equations.

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Since $(4, 4)$ fails the first equation, there is no need to check the second equation.

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Final Answer:

$(4, 4)$ is not a solution to the system of equations.

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Would you like a detailed explanation of how solutions to systems of equations are determined, or do you have any other questions? 😊

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Expanding Questions:

1. What does it mean for a point to be a solution to a system of equations?
2. How can we find all solutions to a system of linear equations?
3. How do the graphs of these two equations intersect?
4. What is the significance of slope in determining solutions?
5. Can a system of equations have no solutions or infinite solutions?

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Tip: When testing if a point satisfies multiple equations, substitute the point into each equation individually to ensure consistency!