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Question 4(Multiple Choice Worth 1 points) (05.03 MC) Determine which system will produce infinitely many solutions.

Ask by Moreno Schofield. in the United States
Jan 24,2025

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Answer

A system has infinitely many solutions if both equations represent the same line.

Solution

To determine which system will produce infinitely many solutions, we need to analyze the equations in the system and see if they are dependent or independent. Let's assume the system of equations is: \[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \] If the two equations are the same, then the system will have infinitely many solutions. This is because the two equations represent the same line, and any point on this line will satisfy both equations. Therefore, the system will produce infinitely many solutions if the two equations are the same.

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The Deep Dive

To produce infinitely many solutions in a system of equations, the equations must represent the same line when graphed. This generally happens when the equations are equivalent, meaning that one can be derived from the other by multiplying or dividing by a non-zero constant. For example, if you have two equations, such as \(2x + 4y = 8\) and \(x + 2y = 4\), they are essentially the same line—which reveals a delightful continuum of solutions! To spot these situations, always look for the coefficients of \(x\) and \(y\) to be proportional across the equations. Common mistakes include miscalculating the coefficients or the constants, leading to false conclusions of no solutions or a unique solution. A quick check by rearranging the equations can aid in accurately identifying infinite solutions!

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