Question

1. solve the equation

\frac{x}{8} = \frac{1}{4} + \frac{x}{6}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = \boxed{-6} (type an integer or a fraction.)
b. there is no solution.

1. solve the equation.

\frac{7}{y} + \frac{1}{2} = \frac{9}{y}
select the correct choice below and fill in any answer boxes present in your choice.
a. the solution is y = \boxed{} (type an integer or a simplified fraction. use a comma to separate answers as n
b. there is no solution.

1. solve the equation.

\frac{x^2 + 5}{x} = \frac{21}{x}

Explanation:

Problem 2: Solve \(\boldsymbol{\frac{x}{8}=\frac{1}{4}+\frac{x}{6}}\)

Step 1: Eliminate denominators

Multiply all terms by the least common multiple (LCM) of 8, 4, and 6, which is 24.
\(24\times\frac{x}{8}=24\times\frac{1}{4}+24\times\frac{x}{6}\)
Simplify each term: \(3x = 6 + 4x\)

Step 2: Solve for \(x\)

Subtract \(3x\) from both sides:
\(3x - 3x = 6 + 4x - 3x\)
Simplify: \(0 = 6 + x\)

Subtract 6 from both sides:
\(x = -6\)

Step 1: Eliminate the variable in denominators

Subtract \(\frac{7}{y}\) from both sides:
\(\frac{7}{y}-\frac{7}{y}+\frac{1}{2}=\frac{9}{y}-\frac{7}{y}\)
Simplify: \(\frac{1}{2}=\frac{2}{y}\)

Step 2: Cross - multiply

Cross - multiply to get: \(y\times1 = 2\times2\)
So, \(y = 4\) (Check: \(\frac{7}{4}+\frac{1}{2}=\frac{7 + 2}{4}=\frac{9}{4}\), and \(\frac{9}{4}=\frac{9}{4}\), valid)

Step 1: Eliminate the denominator

Since \(x
eq0\) (division by zero is undefined), multiply both sides by \(x\):
\(x^{2}+5 = 21\)

Step 2: Solve the quadratic equation

Subtract 21 from both sides:
\(x^{2}+5 - 21 = 0\)
Simplify: \(x^{2}-16 = 0\)

Factor (difference of squares): \((x - 4)(x + 4)=0\)

Set each factor to zero:
\(x - 4 = 0\) or \(x + 4 = 0\)
So, \(x = 4\) or \(x=-4\) (Check: For \(x = 4\), \(\frac{16 + 5}{4}=\frac{21}{4}\); for \(x=-4\), \(\frac{16 + 5}{-4}=\frac{21}{-4}\), both valid)

Answer:

A. \(x = -6\)

Problem 3: Solve \(\boldsymbol{\frac{7}{y}+\frac{1}{2}=\frac{9}{y}}\)