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Which system of equations has a solution of approximately \( (0.7,-1.4) \) ? \( \begin{array}{l}x-y=2 \text { and } x+2 y=-2 \\ x-2 y=-2 \text { and } x+y=2 \\ x-y=2 \text { and } x-2 y=-2 \\ x+y=2 \text { and } x+2 y=-2\end{array} \) Mark this and return

Ask by May Sullivan. in the United States
Mar 12,2025

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Answer

The system \( x - y = 2 \) and \( x + 2y = -2 \) has an approximate solution of \( (0.7, -1.4) \).

Solution

We start by examining each system of equations to see if it has an approximate solution of \( (0.7, -1.4) \). ### Option 1 The system is \[ x - y = 2 \quad \text{and} \quad x + 2y = -2. \] 1. Solve the first equation for \( y \): \[ y = x - 2. \] 2. Substitute into the second equation: \[ x + 2(x - 2) = -2 \quad \Longrightarrow \quad x + 2x - 4 = -2. \] 3. Combine like terms: \[ 3x - 4 = -2. \] 4. Solve for \( x \): \[ 3x = 2 \quad \Longrightarrow \quad x = \frac{2}{3} \approx 0.67. \] 5. Substitute \( x \) back to find \( y \): \[ y = \frac{2}{3} - 2 = -\frac{4}{3} \approx -1.33. \] The approximate solution is \( (0.67, -1.33) \), which is very close to \( (0.7, -1.4) \). ### Option 2 The system is \[ x - 2y = -2 \quad \text{and} \quad x + y = 2. \] 1. Solve the second equation for \( x \): \[ x = 2 - y. \] 2. Substitute into the first equation: \[ (2 - y) - 2y = -2 \quad \Longrightarrow \quad 2 - 3y = -2. \] 3. Solve for \( y \): \[ -3y = -4 \quad \Longrightarrow \quad y \approx 1.33. \] 4. Thus, \( x = 2 - 1.33 \approx 0.67 \). This gives the solution \( (0.67, 1.33) \), which does not match \( (0.7, -1.4) \). ### Option 3 The system is \[ x - y = 2 \quad \text{and} \quad x - 2y = -2. \] 1. From the first equation, \( x = y + 2 \). 2. Substitute into the second: \[ (y + 2) - 2y = -2 \quad \Longrightarrow \quad 2 - y = -2. \] 3. Solve for \( y \): \[ -y = -4 \quad \Longrightarrow \quad y = 4. \] 4. Then, \( x = 4 + 2 = 6 \). The solution here is \( (6, 4) \), which is not an approximate match. ### Option 4 The system is \[ x + y = 2 \quad \text{and} \quad x + 2y = -2. \] 1. Subtract the first equation from the second: \[ (x + 2y) - (x + y) = -2 - 2 \quad \Longrightarrow \quad y = -4. \] 2. Substitute \( y = -4 \) into \( x + y = 2 \): \[ x - 4 = 2 \quad \Longrightarrow \quad x = 6. \] This gives the solution \( (6, -4) \), which is not an approximate match. ### Conclusion Only **Option 1** produces a solution that is approximately \( (0.7, -1.4) \). \[ \boxed{x - y = 2 \quad \text{and} \quad x + 2y = -2.} \]

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Beyond the Answer

Let’s delve into the world of systems of equations! The solution \( (0.7, -1.4) \) indicates that if you plug these values into the equations, they should balance perfectly. A neat way to check is by substituting \( x \) and \( y \) into each pair of equations to see which holds true. It's essential to remember that systems of equations can often have one unique solution, no solution, or infinitely many solutions. So, when you're analyzing systems, verify by substituting values back to confirm the equations align with your solution. Enjoy the quest of finding the equation pair that fits!

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