Solve the following equation, and check the solution. (Hint: Clear the fractions by multiplying each side of the equation by the least common denominator.) $$ -\frac{4}{7} r+4 r=\frac{1}{4} r+\frac{89}{4} $$ The least common denominator (LCD) of $$ \frac{4}{7}, \frac{1}{4} $$, and $$ \frac{89}{4} $$ is $$ \square $$.

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Multiply each term of the equation
$$
-\frac{4}{7}r + 4r = \frac{1}{4}r + \frac{89}{4}
$$
by the least common denominator. The denominators present are $$7$$ and $$4$$, and their least common denominator is
$$
28.
$$

Multiplying every term by $$28$$ gives:
$$
28\left(-\frac{4}{7}r\right) + 28(4r) = 28\left(\frac{1}{4}r\right) + 28\left(\frac{89}{4}\right).
$$

Simplify each term:
$$
28\left(-\frac{4}{7}r\right) = -\frac{28 \times 4}{7}r = -16r,
$$
$$
28(4r) = 112r,
$$
$$
28\left(\frac{1}{4}r\right) = \frac{28}{4}r = 7r,
$$
$$
28\left(\frac{89}{4}\right) = \frac{28 \times 89}{4} = 7 \times 89 = 623.
$$

So the equation becomes:
$$
-16r + 112r = 7r + 623.
$$

Combine like terms on the left side:
$$
96r = 7r + 623.
$$

Subtract $$7r$$ from both sides:
$$
89r = 623.
$$

Divide both sides by $$89$$:
$$
r = \frac{623}{89}.
$$
Since $$89 \times 7 = 623$$, we have:
$$
r = 7.
$$

To check the solution, substitute $$r = 7$$ back into the original equation:
$$
-\frac{4}{7}(7) + 4(7) = -4 + 28 = 24,
$$
$$
\frac{1}{4}(7) + \frac{89}{4} = \frac{7+89}{4} = \frac{96}{4} = 24.
$$

Both sides equal $$24$$, confirming that $$r = 7$$ is the correct solution.

Thus, the least common denominator (LCD) is $$28$$.
The least common denominator (LCD) is 28.

Solve the following equation, and check the solution. (Hint: Clear the fractions by multiplying each side of the equation by the least common denominator.)

The least common denominator (LCD) of , and is .

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Solve the following equation, and check the solution. (Hint: Clear the fractions by multiplying each side of the equation by the least common denominator.) The least common denominator (LCD) of , and is .