To find the solution to the system of equations represented by the two intersecting lines, we first need to determine the equations of both lines based on the points provided.
Line 1
The first line passes through the points (−3, −4), (0, −2), and (3, 0).
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We can find the slope (m) using the points (0, -2) and (3, 0): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{3 - 0} = \frac{2}{3} \]
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Now, we can use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\) with point (0, -2): \[ y - (-2) = \frac{2}{3}(x - 0) \implies y + 2 = \frac{2}{3}x \implies y = \frac{2}{3}x - 2 \]
Line 2
The second line passes through the points (−3, −4), (−1, 0), (0, 2), and (1, 4).
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We can find the slope (m) using the points (−3, -4) and (−1, 0): \[ m = \frac{0 - (-4)}{-1 - (-3)} = \frac{4}{2} = 2 \]
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Using the point-slope form with the point (-3, -4): \[ y - (-4) = 2(x + 3) \implies y + 4 = 2x + 6 \implies y = 2x + 2 \]
Finding the Intersection
Now, we have two equations:
- \(y = \frac{2}{3}x - 2\) (Line 1)
- \(y = 2x + 2\) (Line 2)
To find the intersection, we set the two equations equal: \[ \frac{2}{3}x - 2 = 2x + 2 \]
To eliminate the fraction, multiply through by 3: \[ 2x - 6 = 6x + 6 \]
Now, move all terms involving x to one side and constant terms to the other: \[ 2x - 6x = 6 + 6 \implies -4x = 12 \implies x = -3 \]
Now substitute \(x = -3\) back into either equation to find \(y\): Using Line 1: \[ y = \frac{2}{3}(-3) - 2 = -2 - 2 = -4 \]
Thus the solution to the system of equations is: \[ (-3, -4) \]
The coordinates are:
- \(x = -3\)
- \(y = -4\)
Final answer: (-3, -4)