proyaop

Let S = 0.296296296296...

1000S = 296.296296296... = S + 296

999S = 296

S = 296/999 = (37)(8)/(37)(27) = 8/27 

\(\frac{1}{64a^3 + 7} - 7 = 0\)

Add 7 to both sides: \(\frac{1}{64a^3 + 7} =7\)

Multiply both sides by 64a^3 + 7 to get rid of denominator: \(1 = 7(64a^3+7)\); \(1 = 448a^3+49\); \(-48 =448a^3\); \(-{3\over28}=a^3\)

Cube rooting both sides, you get \(a = ({-{3\over28}})^{1\over3}\), or noted as the cube root of -3/28.

\(\frac{a}{3} + 1 = \frac{a + 3}{a} + 1\)

Subtract 1 from both sides: \({a\over3}={a+3\over{a}}\)

Cross multiply: \(a^2=3a+9\); \(a^2-3a-9=0\)

Use quadratic formula: \(a = {3 \pm\sqrt{9+36}\over2}={3\pm3\sqrt{5}\over2}\)

t = cost of toast, b = cost of bagel, m = cost of muffin

2t + b = 3.15                              Equation 1

b + m = 3.5                                 Equation 2

t + 2b + 3m = 8.15                      Equation 3

Use elimination:

2*Equation 3 - Equation 1:

2t + 4b + 6m - 2t - b = 16.3 - 3.15; 3b + 6m = 13.15                      Equation 3.1

6*Equation2 - Equation 3.1:

6b + 6m - 3b - 6m = 21 - 13.15

3b = 7.85

b = 261.666666666... cents

So, a bagel costs (roughly) 262 cents.

Add the two equations together and you get 2a = 10, a = 5, which is your only solution.

p + r = 9 + p, subtract p from both sides, you get r = 9.

q - 2r = -2 + q, subtract q from both sides, you get -2r = -2, r = 1. 

1 is not equal to 9, so there may be a latex or copy and paste issue when transferring websites 

\(ab^4=48\), \(ab^2=4\)

\(ab^4=(ab^2)b^2=4b^2=48\)

\(b^2=12\), \(b=\pm2\sqrt{3}\)

If b has to be positive as specified possibly in your question, then type 2sqrt(3), if not, then type -2sqrt(3).

Interior angle sum of ABC is 180 degrees, so p + 2q + 6p - 5q + 14p + 8q = 180

Combine like terms: 21p + 5q = 180. 

21p = 180 - 5q

p = (180 - 5q)/21