Bot GPT 4-o mini To solve the system of equations given by \( x - 5y = 7 \) and \( 3x - 2y = -4 \), we can use either the substitution method or the elimination method. Here, I will demonstrate the substitution method:
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Rearranging the First Equation: Start with the first equation \( x - 5y = 7 \). Rearranging gives us: \[ x = 5y + 7 \]
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Substituting into the Second Equation: Substitute \( x \) in the second equation \( 3x - 2y = -4 \): \[ 3(5y + 7) - 2y = -4 \] Simplifying this, we have: \[ 15y + 21 - 2y = -4 \] Combining like terms: \[ 13y + 21 = -4 \]
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Isolating \( y \): Now, isolate \( y \): \[ 13y = -4 - 21 \] \[ 13y = -25 \] \[ y = -\frac{25}{13} \]
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Substituting \( y \) Back to Find \( x \): Now substitute \( y \) back into the expression for \( x \): \[ x = 5\left(-\frac{25}{13}\right) + 7 \] \[ x = -\frac{125}{13} + 7 \] \[ x = -\frac{125}{13} + \frac{91}{13} \] \[ x = -\frac{125 - 91}{13} \] \[ x = -\frac{34}{13} \]
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Final Solution: Thus, the intersection point, which is the solution to the system of equations, is: \[ \left(-\frac{34}{13}, -\frac{25}{13}\right) \]
So, the intersection point is \(\left(-\frac{34}{13}, -\frac{25}{13}\right)\).
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