Solving a Linear Equation with Fractions
The equation provided in the image is:
\[ \frac{2}{3} - x + \frac{1}{2} = \frac{x}{2} - 1 \]
Let's solve for \( x \).
First, we want to combine like terms and get all the \( x \) terms on one side of the equation and the constants on the other side. To do this, it will be easier to work with a common denominator for the fractions. The smallest common denominator for 3 and 2 is 6.
Let's convert each fraction to have denominator 6:
\[ \frac{2}{3} \cdot \frac{2}{2} = \frac{4}{6} \]
\[ \frac{1}{2} \cdot \frac{3}{3} = \frac{3}{6} \]
\[ \frac{x}{2} \cdot \frac{3}{3} = \frac{3x}{6} \]
Now rewrite the equation with these equivalent fractions:
\[ \frac{4}{6} - x + \frac{3}{6} = \frac{3x}{6} - 1 \]
Combine the constant fractions on the left side:
\[ \frac{4}{6} + \frac{3}{6} = \frac{7}{6} \]
Now the equation looks like this:
\[ \frac{7}{6} - x = \frac{3x}{6} - 1 \]
Add \( x \) to both sides to get all \( x \) terms on the right side:
\[ \frac{7}{6} = \frac{3x}{6} + x - 1 \]
Recognize that \( x \) is the same as \( \frac{6x}{6} \) and add it to the \( \frac{3x}{6} \):
\[ \frac{7}{6} = \frac{3x}{6} + \frac{6x}{6} - 1 \]
Combine the \( x \) terms:
\[ \frac{7}{6} = \frac{9x}{6} - 1 \]
Now we'll solve for \( x \) by isolating it on one side. First, add 1 to both sides to move the constant term to the left side:
\[ \frac{7}{6} + 1 = \frac{9x}{6} \]
Convert 1 to a fraction with a denominator of 6 to combine it with the \( \frac{7}{6} \):
\[ \frac{7}{6} + \frac{6}{6} = \frac{9x}{6} \]
Combine the fractions on the left side:
\[ \frac{13}{6} = \frac{9x}{6} \]
Multiply both sides by \( \frac{6}{9} \) (which is the reciprocal of the coefficient of \( x \)) to isolate \( x \):
\[ \frac{13}{6} \cdot \frac{6}{9} = \frac{9x}{6} \cdot \frac{6}{9} \]
Simplify:
\[ \frac{13}{9} = x \]
So, the solution to the equation is:
\[ x = \frac{13}{9} \]
