Question

identify the number of solutions to the linear equation given below. 0.25x + \frac{2}{5}x = 0.5x - \frac{9}{30} answer:

Explanation:

Step1: Convert fractions and decimals

First, convert \(0.25=\frac{1}{4}\), \(0.5 = \frac{1}{2}\), and \(\frac{9}{30}=\frac{3}{10}\). The equation becomes \(\frac{1}{4}x+\frac{2}{5}x=\frac{1}{2}x - \frac{3}{10}\).

Step2: Find a common - denominator

The common denominator of 4, 5, and 10 is 20. Rewrite the left - hand side and right - hand side: \(\frac{1\times5}{4\times5}x+\frac{2\times4}{5\times4}x=\frac{1\times10}{2\times10}x-\frac{3\times2}{10\times2}\), which simplifies to \(\frac{5}{20}x+\frac{8}{20}x=\frac{10}{20}x-\frac{6}{20}\).

Step3: Combine like terms

On the left - hand side, \(\frac{5 + 8}{20}x=\frac{13}{20}x\). The equation is now \(\frac{13}{20}x=\frac{10}{20}x-\frac{6}{20}\).

Step4: Move \(x\) terms to one side

Subtract \(\frac{10}{20}x\) from both sides: \(\frac{13}{20}x-\frac{10}{20}x=-\frac{6}{20}\).

Step5: Simplify

\(\frac{13 - 10}{20}x=-\frac{6}{20}\), so \(\frac{3}{20}x=-\frac{6}{20}\).

Step6: Solve for \(x\)

Multiply both sides by \(\frac{20}{3}\), we get \(x = - 2\). Since we have found a single value for \(x\), there is 1 solution.

Answer:

1