Question - Solving a Linear Equation with Fractions
mathsolving linear equationisolating variable in equationlinear equation fractionscombining fractions with variables
Solution:
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:\[ x = \frac{13}{9} \]
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