Question

bonus 16. find ml n 110 + 4x x + 103 70 m l k x + 243 bonus 17. find ge h 13x - 993 4 g f e 3x - 129 3x - 185

Explanation:

Step1: Set up equation for the first line - segment problem

Since $NL=NK$, we have $(110 + 4x)+(x + 103)=x + 243$.
First, simplify the left - hand side:
$110+4x+x + 103=5x+213$.
So the equation becomes $5x+213=x + 243$.

Step2: Solve the equation for $x$

Subtract $x$ from both sides: $5x - x+213=x - x + 243$, which simplifies to $4x+213=243$.
Then subtract 213 from both sides: $4x+213 - 213=243 - 213$, so $4x = 30$, and $x=\frac{30}{4}=7.5$.

Step3: Find $ML$

Substitute $x = 7.5$ into the expression for $ML$ which is $x + 103$.
$ML=7.5+103=110.5$.

Step4: Set up equation for the second line - segment problem

Since $HG+GE=HE$, we have $(3x - 129)+(3x - 185 + 4)=13x - 993$.
Simplify the left - hand side: $3x-129+3x-181 = 6x-310$.
So the equation is $6x-310=13x - 993$.

Step5: Solve the equation for $x$

Subtract $6x$ from both sides: $6x-6x-310=13x-6x - 993$, which gives $-310 = 7x-993$.
Add 993 to both sides: $-310 + 993=7x-993 + 993$, so $683=7x$.
Then $x=\frac{683}{7}\approx97.57$.

Step6: Find $GE$

$GE=3x-185 + 4=3x-181$.
Substitute $x=\frac{683}{7}$ into the expression for $GE$:
$GE=3\times\frac{683}{7}-181=\frac{2049}{7}-\frac{1267}{7}=\frac{2049 - 1267}{7}=\frac{782}{7}\approx111.71$.

Answer:

$ML = 110.5$, $GE=\frac{782}{7}\approx111.71$