To find the solution for the equations \( y = -\frac{1}{2}x - 6 \) and \( y = \frac{7}{6}x + 4 \), we will first graph the two lines and then find their intersection point, which represents the solution to the system of equations.
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Graphing the Functions:
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Equation 1: \( y = -\frac{1}{2}x - 6 \)
- This is a linear equation. The slope (m) is \(-\frac{1}{2}\) and the y-intercept (b) is -6.
- You can plot the y-intercept (0, -6) and then use the slope to find another point. From (0, -6), moving down 1 unit and right 2 units gives you the point (2, -7).
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Equation 2: \( y = \frac{7}{6}x + 4 \)
- This line has a slope of \(\frac{7}{6}\) and a y-intercept of 4.
- Start at the y-intercept (0, 4) and use the slope to find another point. From (0, 4), moving up 7 units and right 6 units gives you the point (6, 11).
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Finding the Intersection Point:
To find the intersection point, set the equations equal to each other:
\[ -\frac{1}{2}x - 6 = \frac{7}{6}x + 4 \]
To eliminate the fractions, multiply the entire equation by 6:
\[ 6 \left(-\frac{1}{2}x - 6\right) = 6 \left(\frac{7}{6}x + 4\right) \] \[ -3x - 36 = 7x + 24 \]
Now, combine like terms:
\[ -3x - 7x = 24 + 36 \] \[ -10x = 60 \] \[ x = -6 \]
Now substitute \( x = -6 \) back into either original equation to find \( y \):
Using the first equation: \[ y = -\frac{1}{2}(-6) - 6 \] \[ y = 3 - 6 = -3 \]
So the intersection point (solution) is: \[ (x, y) = (-6, -3) \]
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Final Result:
The solution to the system of equations \( y = -\frac{1}{2}x - 6 \) and \( y = \frac{7}{6}x + 4 \) is the point: \[ \boxed{(-6, -3)} \]