To solve the equation [tex]\(\frac{15(5-x)-7(x+9)}{1-5 x}=4\)[/tex], let's go through it step-by-step.
1. Simplify the numerator:
[tex]\[
15(5 - x) - 7(x + 9)
\][/tex]
Distribute the 15 and -7:
[tex]\[
15 \cdot 5 - 15x - 7x - 7 \cdot 9
\][/tex]
Simplify the terms:
[tex]\[
75 - 15x - 7x - 63
\][/tex]
Combine like terms:
[tex]\[
75 - 63 - 15x - 7x = 12 - 22x
\][/tex]
2. Rewrite the original equation with the simplified numerator:
[tex]\[
\frac{12 - 22x}{1 - 5x} = 4
\][/tex]
3. Multiply both sides by the denominator to clear the fraction:
[tex]\[
12 - 22x = 4(1 - 5x)
\][/tex]
4. Distribute the 4 on the right side of the equation:
[tex]\[
12 - 22x = 4 - 20x
\][/tex]
5. Get all the [tex]\(x\)[/tex]-terms on one side and the constant terms on the other:
Add [tex]\(20x\)[/tex] to both sides:
[tex]\[
12 - 22x + 20x = 4
\][/tex]
Simplify:
[tex]\[
12 - 2x = 4
\][/tex]
Subtract 12 from both sides:
[tex]\[
-2x = 4 - 12
\][/tex]
Simplify the right-hand side:
[tex]\[
-2x = -8
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{-8}{-2}
\][/tex]
Simplify:
[tex]\[
x = 4
\][/tex]
Thus, the solution to the equation [tex]\(\frac{15(5-x)-7(x+9)}{1-5 x}=4\)[/tex] is [tex]\(\boxed{4}\)[/tex].
