To understand the solution to the system of equations:
[tex]\[ y = -\frac{1}{2} x + 9 \][/tex]
[tex]\[ y = x + 7 \][/tex]
we need to determine the conditions under which these two lines intersect.
1. First equation analysis:
The equation [tex]\( y = -\frac{1}{2} x + 9 \)[/tex] represents a line with a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept of [tex]\(9\)[/tex].
2. Second equation analysis:
The equation [tex]\( y = x + 7 \)[/tex] represents a line with a slope of [tex]\(1\)[/tex] and a y-intercept of [tex]\(7\)[/tex].
To find the intersection point of these two lines, we set the expressions for [tex]\( y \)[/tex] equal to each other because at the intersection point, both [tex]\( y \)[/tex]-values will be the same:
[tex]\[ -\frac{1}{2} x + 9 = x + 7 \][/tex]
Solving this equation for [tex]\( x \)[/tex]:
1. Combine like terms:
[tex]\[ 9 - 7 = x + \frac{1}{2} x \][/tex]
[tex]\[ 2 = 1.5x \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2}{1.5} \][/tex]
[tex]\[ x = \frac{4}{3} \][/tex]
Thus, the [tex]\( x \)[/tex]-coordinate of the intersection point is [tex]\(\frac{4}{3}\)[/tex].
Next, we substitute [tex]\( x = \frac{4}{3} \)[/tex] back into either of the original equations to find the [tex]\( y \)[/tex]-coordinate. Using [tex]\( y = x + 7 \)[/tex]:
[tex]\[ y = \frac{4}{3} + 7 \][/tex]
[tex]\[ y = \frac{4}{3} + \frac{21}{3} \][/tex]
[tex]\[ y = \frac{25}{3} \][/tex]
Therefore, the intersection point is [tex]\(\left( \frac{4}{3}, \frac{25}{3} \right)\)[/tex].
So the correct description of the solution to the given system of equations is:
Line [tex]\( y = -\frac{1}{2} x + 9 \)[/tex] intersects line [tex]\( y = x + 7 \)[/tex].
