To determine whether the system of equations defined by the points (3, 10) and (0, 1) has more than one solution, we first need to establish a relationship between these points.
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Identify the equation of the line: We can derive the linear equation that passes through the points (3, 10) and (0, 1).
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Find the slope (m): The slope \( m \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3. \]
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Use point-slope form: Now that we have the slope, we can use one of the points (let's use (0, 1)) to find the line equation in point-slope form, which is: \[ y - y_1 = m(x - x_1). \] Plugging in (0, 1): \[ y - 1 = 3(x - 0) \implies y - 1 = 3x \implies y = 3x + 1. \]
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Verify the other point: Now we check if the other point (3, 10) satisfies this equation: \[ y = 3(3) + 1 = 9 + 1 = 10. \] Since (3, 10) lies on the line defined by \( y = 3x + 1 \), this confirms that both points define the same linear equation.
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Does the system have more than one solution?: The equation \( y = 3x + 1 \) represents an infinite number of solutions because it is a linear equation. Any value of \( x \) will produce a corresponding value of \( y \) on the line. Therefore, there are indeed more than one solution to the system, as any point satisfying this linear equation is a solution.
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Conclusion: Thus, the system of equations, defined by the points (3, 10) and (0, 1), has an infinite number of solutions represented by the line \( y = 3x + 1 \).
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