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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{r}5 x+4 y=4 \\ -10 x-8 y=-8\end{array} \] Answer One Solution Infinitely Many Solutions

Ask by Norton Li. in the United States
Jan 22,2025

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The system has infinitely many solutions.

Solution

To determine whether the given system of equations has no solutions, infinitely many solutions, or exactly one solution, let's analyze the system: \[ \begin{cases} 5x + 4y = 4 \quad \text{(Equation 1)} \\ -10x - 8y = -8 \quad \text{(Equation 2)} \end{cases} \] **Step 1: Check for Proportionality** First, let's see if Equation 2 is a multiple of Equation 1. Multiply Equation 1 by **-2**: \[ -2(5x + 4y) = -2(4) \\ -10x - 8y = -8 \] This resulting equation is exactly the same as Equation 2: \[ -10x - 8y = -8 \] **Step 2: Interpretation** Since Equation 2 is just a multiple of Equation 1, both equations represent the **same line** in the coordinate plane. This means that every solution to Equation 1 is also a solution to Equation 2, and vice versa. **Conclusion:** Because both equations describe the same line, there are **infinitely many solutions** to the system. **Answer:** Infinitely Many Solutions

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Mind Expander

The given system of equations can be manipulated to determine its solution set. If we multiply the first equation \(5x + 4y = 4\) by \(-2\), we end up with \(-10x - 8y = -8\), which is exactly the second equation. This indicates that both equations represent the same line in the coordinate system. Therefore, the two equations have infinitely many solutions that lie on this line. To visualize the relationship between these two equations, you can graph both on the same coordinate plane. Since both of them overlap entirely, you'll see that they are not just intersecting at one point - instead, every point on that line is a solution! This means that there's no unique solution, but rather an entire set of solutions to choose from.

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