To solve the equation \( y = -\frac{1}{2}x - 6 \), we'll first graph the equation and then find the solution points.
Step 1: Identify the components for graphing
The equation is in slope-intercept form \( y = mx + b \), where:
- \( m \) is the slope, which is \( -\frac{1}{2} \).
- \( b \) is the y-intercept, which is \( -6 \).
Step 2: Plot the y-intercept
Start by plotting the y-intercept:
- The y-intercept is at \( (0, -6) \).
Step 3: Use the slope to find another point
From the y-intercept, apply the slope:
- The slope \( -\frac{1}{2} \) means that for each 2 units you move to the right (positive x-direction), you move down 1 unit (negative y-direction).
Starting from the point \( (0, -6) \):
- Move 2 units to the right to \( (2, -6) \).
- Move 1 unit down to \( (2, -7) \).
Thus, the second point is \( (2, -7) \).
Step 4: Draw the graph
Plot both points \( (0, -6) \) and \( (2, -7) \) on a graph and draw a straight line through them. This line represents the equation \( y = -\frac{1}{2}x - 6 \).
Step 5: Finding solutions
Since the equation describes a line, any point \( (x, y) \) that lies on this line is a solution to the equation.
If you want specific values of \( x \) and \( y \), you can choose any \( x \) value and solve for \( y \):
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Example 1: If \( x = 0 \): \[ y = -\frac{1}{2}(0) - 6 = -6 \] So, \( (0, -6) \) is a solution.
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Example 2: If \( x = 2 \): \[ y = -\frac{1}{2}(2) - 6 = -1 - 6 = -7 \] So, \( (2, -7) \) is a solution.
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Example 3: If \( x = 4 \): \[ y = -\frac{1}{2}(4) - 6 = -2 - 6 = -8 \] So, \( (4, -8) \) is another solution.
Conclusion
The graph is a straight line with a negative slope and a y-intercept at \( -6 \). Any point on this line is a solution to the equation \( y = -\frac{1}{2}x - 6 \).
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