To describe the solution set for the system of equations consisting of a line and the equation \( 2x + y = 6 \), you first need to express the line's equation and analyze its intersection with the line defined by \( 2x + y = 6 \).
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Identify the Line's Equation: Suppose the equation of the line is given. If you have a specific equation in mind, please provide it for a more precise solution. However, for the sake of this explanation, let's say the line's equation is defined as \( y = mx + b \).
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Set Both Equations Equal to Find Intersection: We want to find where the line intersects with the line represented by \( 2x + y = 6 \). To do that, we can substitute \( y \) from the line's equation into \( 2x + y = 6 \).
If the line's equation is \( y = mx + b \): \[ 2x + (mx + b) = 6. \]
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Solve for \( x \): Combine like terms: \[ (2 + m)x + b = 6. \] So, \[ (2 + m)x = 6 - b. \] This gives: \[ x = \frac{6 - b}{2 + m} \text{ (if \( 2 + m \neq 0 \))}. \]
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Find \( y \): Substitute \( x \) back into the line's equation to find \( y \): \[ y = m\left(\frac{6 - b}{2 + m}\right) + b. \]
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Write the Ordered Pair: The intersection point (the solution to the system) can be expressed as the ordered pair \( (x, y) \).
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Special Cases: If the lines are parallel (i.e., have the same slope and different y-intercepts), there will be no solution (the lines do not intersect). If they are the same line (i.e., all points satisfy both equations), the solution set would include all points on that line.
If you provide the specific equation of the line, I can give you the exact coordinates of the intersection point.
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