Ok, so
Here we have the following segments:
We want to find the value of the measure of GE.
Notice that we have that:
[tex]\begin{gathered} HG=13x-993 \\ HF=3x-129 \\ GE=3x-185 \\ FE=4 \end{gathered}[/tex]
If we look at the graph, we can see that the measure of GF can be found if we substract HF and HG. Right?
Therefore, GF will be:
[tex]\begin{gathered} GF=HF-HG \\ GF=3x-129-(13x-993) \end{gathered}[/tex]
Multiplying the sign per the terms inside the brackets, we got that:
[tex]\begin{gathered} GF=3x-129-13x+993 \\ GF=-10x+864 \end{gathered}[/tex]
And, the segment GE can be found if we sum GF and FE.
This is:
[tex]\begin{gathered} GE=GF+FE \\ GE=-10x+864+4 \\ GE=-10x+868 \end{gathered}[/tex]
Now, we can see that GE is also given by the equation 3x - 185. So, we could equal both equations:
[tex]\begin{gathered} GE=GE \\ -10x+868=3x-185 \end{gathered}[/tex]
And if we solve this equation for x:
[tex]\begin{gathered} -10x-3x=-185-868 \\ -13x=-1053 \\ x=\frac{-1053}{-13}=81 \end{gathered}[/tex]
Now, we know that x=81.
To find the measure of GE, we just replace x=81 in any of both equations given for GE. We could replace in the second one because it is easier.
[tex]GE=3x-185=3(81)-185=58[/tex]
Therefore,
[tex]GE=58[/tex]
