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Transcription:

The image shows the following problem:

Consider the following system of linear equations:

mx + 4y &= 3 \\ 3x + my &= 4 \end{align*}

For which of the following values of $m$ does the system have infinite solutions?

a) $m = 4$
b) $m = -3$
c) $m = 3, 4$
d) $m = -1$

Solution:

For a system of linear equations to have infinite solutions, the two equations must be multiples of each other. In other words, they should be linearly dependent.

Let's check this condition for the given system:

1. Original Equations: $\text{Equation 1: } mx + 4y = 3$ $\text{Equation 2: } 3x + my = 4$

2. Check for Linear Dependence: The equations are dependent if: $\frac{m}{3} = \frac{4}{m} = \frac{3}{4}$ For the first and second ratios to be equal, we must have: $\frac{m}{3} = \frac{3}{4} \quad \text{or} \quad m = 3$ But, substituting this into the second equation does not make both equations equal, hence no linear dependence.

3. Check with the given options:

   • For $m = 4$: $\text{Equation 1: } 4x + 4y = 3$ $\text{Equation 2: } 3x + 4y = 4$ These are not multiples of each other.

   • For $m = -3$: $\text{Equation 1: } -3x + 4y = 3$ $\text{Equation 2: } 3x - 3y = 4$ These are not multiples of each other.

   • For $m = -1$: $\text{Equation 1: } -x + 4y = 3$ $\text{Equation 2: } 3x - y = 4$ These are not multiples of each other.

   • Therefore, there is no value of $m$ among the given options that make the system have infinite solutions. The system has a unique solution for most values of $m$, and it becomes inconsistent (no solution) for some specific values of $m$.

Conclusion:

None of the provided options are correct for infinite solutions based on the reasoning above.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

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4. What are the conditions for a system of linear equations to have a unique solution?
5. How can you solve a system of equations using matrices?
6. What is the determinant, and how does it relate to solutions of linear systems?
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8. What are the differences between homogeneous and non-homogeneous linear systems?

Tip: In solving linear equations, understanding the conditions for unique, infinite, and no solutions helps in quickly analyzing the system without having to solve it fully each time.