To determine how many solutions the system of equations has, we need to analyze the two equations provided:
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The first equation is in slope-intercept form: \[ y = -2x + 12 \] Here, the slope is -2 and the y-intercept is 12.
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The second equation can be rearranged into slope-intercept form as well. Starting with: \[ 4y + x - 20 = 0 \] we isolate \(y\): \[ 4y = -x + 20 \] \[ y = -\frac{1}{4}x + 5 \] In this equation, the slope is \(-\frac{1}{4}\) and the y-intercept is 5.
Now, we can analyze the slopes of the two lines:
- The slope of the first line is -2.
- The slope of the second line is \(-\frac{1}{4}\).
Since the slopes of the two lines are different, the two lines are not parallel and will intersect at one point.
Thus, the system of equations has one solution.