Certainly! Let's solve the given equations step by step.
Given equations:
1. [tex]\(\frac{3}{4}(e-1)\)[/tex]
2. [tex]\(\frac{5}{3}(e-8)\)[/tex]
3. [tex]\(\frac{8}{3}(6-k) \times \frac{5}{12}\)[/tex]
We need to determine the values of [tex]\(e\)[/tex] and [tex]\(k\)[/tex] that satisfy these equations. Let's start by setting the first two expressions equal to each other:
[tex]\[
\frac{3}{4}(e-1) = \frac{5}{3}(e-8)
\][/tex]
### Solving for [tex]\(e\)[/tex]:
1. Cross-multiply to clear the fractions:
[tex]\[
3 \times 3(e-1) = 4 \times 5(e-8)
\][/tex]
[tex]\[
9(e-1) = 20(e-8)
\][/tex]
2. Distribute and simplify:
[tex]\[
9e - 9 = 20e - 160
\][/tex]
3. Isolate [tex]\(e\)[/tex]:
[tex]\[
9e - 20e = -160 + 9
\][/tex]
[tex]\[
-11e = -151
\][/tex]
[tex]\[
e = \frac{151}{11}
\][/tex]
[tex]\[
e =13.727... \approx 13.727
\][/tex]
Now let's relate the second and third expressions:
[tex]\[
\frac{5}{3}(e-8) = \frac{8}{3}(6-k) \times \frac{5}{12}
\][/tex]
### Simplify the third expression:
1. Simplify the fraction inside the third expression:
[tex]\[
\frac{8}{3}(6-k) \times \frac{5}{12} = \frac{8 \times 5}{3 \times 12}(6-k) = \frac{40}{36}(6-k) = \frac{10}{9}(6-k)
\][/tex]
So, we have:
[tex]\[
\frac{5}{3}(e-8) = \frac{10}{9}(6-k)
\][/tex]
2. Cross-multiply again to clear the fractions:
[tex]\[
5 \times 9(e-8) = 3 \times 10(6-k)
\][/tex]
[tex]\[
45(e-8) = 30(6-k)
\][/tex]
3. Distribute and simplify:
[tex]\[
45e - 360 = 180 - 30k
\][/tex]
4. Isolate [tex]\(k\)[/tex]:
[tex]\[
45e - 360 - 180 = -30k
\][/tex]
[tex]\[
45e - 540 = -30k
\][/tex]
[tex]\[
30k = 540 - 45e
\][/tex]
[tex]\[
k = \frac{540 - 45e}{30}
\][/tex]
Now, substitute the value of [tex]\(e\)[/tex] into the equation:
[tex]\[
e = \frac{151}{11}
\][/tex]
5. Substitute [tex]\(e\)[/tex] and solve for [tex]\(k\)[/tex]:
1.
[tex]\[
k = \frac{540 - 45 \times \frac{151}{11}}{30}
\][/tex]
2.
[tex]\[
k = \frac{540 - \frac{6795}{11}}{30}
\][/tex]
[tex]\[
k = \frac{540 \times 11 - 6795}{30 \times 11}
\][/tex]
[tex]\[
k = \frac{5940 - 6795}{330}
\][/tex]
[tex]\[
k = \frac{-855}{330}
\][/tex]
[tex]\[
k \approx -2.591
\][/tex]
Hence, the solutions are:
[tex]\[
e \approx 13.727,\quad k \approx -2.591
\][/tex]
