Question

for problems 3–12, find the number of solutions to the equation. if the equation has only one solution, solve the equation and check your solution.

1. (\frac{1}{5}h - 3 = 2 + \frac{1}{5}h)
2. (w = \frac{1}{3}w + \frac{2}{3})

Explanation:

Problem 3: Solve \(\boldsymbol{\frac{1}{5}h - 3 = 2 + \frac{1}{5}h}\) and check the solution.

Step 1: Subtract \(\frac{1}{5}h\) from both sides

To eliminate the \(h\)-terms, we subtract \(\frac{1}{5}h\) from each side of the equation.
\(\frac{1}{5}h - 3 - \frac{1}{5}h = 2 + \frac{1}{5}h - \frac{1}{5}h\)
Simplifying both sides:
\(-3 = 2\)

Step 1: Subtract \(\frac{1}{3}w\) from both sides

To isolate the \(w\)-terms on one side, subtract \(\frac{1}{3}w\) from both sides.
\(w - \frac{1}{3}w = \frac{1}{3}w + \frac{2}{3} - \frac{1}{3}w\)
Simplify the left side (using \(w = \frac{3}{3}w\)):
\(\frac{3}{3}w - \frac{1}{3}w = \frac{2}{3}\)
\(\frac{2}{3}w = \frac{2}{3}\)

Step 2: Multiply both sides by \(\frac{3}{2}\)

To solve for \(w\), multiply both sides by the reciprocal of \(\frac{2}{3}\) (which is \(\frac{3}{2}\)):
\(\frac{3}{2} \cdot \frac{2}{3}w = \frac{3}{2} \cdot \frac{2}{3}\)
Simplifying both sides:
\(w = 1\)

Step 3: Check the solution

Substitute \(w = 1\) back into the original equation:
Left side: \(w = 1\)
Right side: \(\frac{1}{3}(1) + \frac{2}{3} = \frac{1}{3} + \frac{2}{3} = 1\)

Since the left side (\(1\)) equals the right side (\(1\)), \(w = 1\) is a valid solution.

Answer:

(for Problem 3):
The equation \(\frac{1}{5}h - 3 = 2 + \frac{1}{5}h\) simplifies to \(-3 = 2\), which is a contradiction. Therefore, the equation has no solution.

Problem 4: Solve \(\boldsymbol{w = \frac{1}{3}w + \frac{2}{3}}\) and check the solution.