Let's solve the equation [tex]\(\frac{90 + 11b}{99} = 0.3(a + b)\)[/tex] step-by-step:
1. Equation Setup: We start with the given equation:
[tex]\[
\frac{90 + 11b}{99} = 0.3(a + b)
\][/tex]
2. Clear the Fraction: To eliminate the fraction, multiply both sides of the equation by 99:
[tex]\[
90 + 11b = 99 \times 0.3(a + b)
\][/tex]
3. Simplify the Right Side: Calculate [tex]\(99 \times 0.3\)[/tex]:
[tex]\[
99 \times 0.3 = 29.7
\][/tex]
Thus, the equation simplifies to:
[tex]\[
90 + 11b = 29.7(a + b)
\][/tex]
4. Distribute on the Right Side: Distribute [tex]\(29.7\)[/tex] across [tex]\(a + b\)[/tex]:
[tex]\[
90 + 11b = 29.7a + 29.7b
\][/tex]
5. Rearrange the Equation: Move all terms involving [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to one side of the equation to isolate [tex]\(a\)[/tex]:
[tex]\[
90 + 11b - 29.7b = 29.7a
\][/tex]
6. Combine Like Terms: Combine the [tex]\(b\)[/tex] terms on the left side:
[tex]\[
90 + (11b - 29.7b) = 29.7a
\][/tex]
[tex]\[
90 - 18.7b = 29.7a
\][/tex]
7. Solve for [tex]\(a\)[/tex]: Isolate [tex]\(a\)[/tex] by dividing both sides by 29.7:
[tex]\[
a = \frac{90 - 18.7b}{29.7}
\][/tex]
8. Express [tex]\(a\)[/tex] in Simplified Form: Simplify the fraction:
[tex]\[
a = 3.03030303030303 - 0.62962962962963b
\][/tex]
So, the solution to the equation [tex]\(\frac{90 + 11b}{99} = 0.3(a + b)\)[/tex] is given by:
[tex]\[
a = 3.03030303030303 - 0.62962962962963b, \quad \text{for any value of } b.
\][/tex]
