Question
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\[ \begin{array}{l}8 x-6 y=5 \\ 4 x-3 y=3\end{array} \] Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The solution of the system is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.

Ask by Morgan Bryant. in the United States
Mar 10,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

There is no solution.

Solution

We start with the system \[ \begin{aligned} 8x - 6y &= 5, \\ 4x - 3y &= 3. \end{aligned} \] Notice that multiplying the second equation by \(2\) gives: \[ 2(4x - 3y) = 2 \cdot 3 \quad \Longrightarrow \quad 8x - 6y = 6. \] Now we have two equations: \[ 8x - 6y = 5 \quad \text{and} \quad 8x - 6y = 6. \] Clearly, \(8x - 6y\) cannot be both \(5\) and \(6\) at the same time. Therefore, the system is inconsistent and there is no solution. Thus, the correct choice is: C. There is no solution.

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Bonus Knowledge

To solve the system of equations \(8x - 6y = 5\) and \(4x - 3y = 3\), we can first notice that the second equation is essentially half of the first equation. If we multiply the second equation by 2, we get: \[ 2(4x - 3y) = 2(3) \implies 8x - 6y = 6 \] Now we have: 1. \(8x - 6y = 5\) 2. \(8x - 6y = 6\) These two equations cannot both be true simultaneously, as they imply that \(5 = 6\), which is a contradiction. Therefore, the correct choice is C. There is no solution.

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