QuestionSolve the equation: $\frac{1}{2} x+\frac{3}{7}=\frac{2}{3}+\frac{8}{21} x$

Studdy Solution

STEP 1

Assumptions1. We are solving for the variable $x$ in the equation $\frac{1}{} x+\frac{3}{7}=\frac{}{3}+\frac{8}{21} x$

STEP 2

First, we want to gather all terms with $x$ on one side of the equation and the constants on the other side. Let's start by subtracting $\frac{8}{21} x$ from both sides of the equation.
$\frac{1}{2} x - \frac{8}{21} x = \frac{2}{} - \frac{}{7}$

STEP 3

Next, we need to subtract fractions. To do this, we need to find a common denominator. The least common multiple of2 and21 is42, so we multiply the numerator and denominator of $\frac{1}{2}$ by21 and the numerator and denominator of $\frac{8}{21}$ by2.
$\frac{21}{42} x - \frac{16}{42} x = \frac{2}{3} - \frac{3}{7}$

STEP 4

Now, subtract the fractions on the left side of the equation.
$\frac{}{42} x = \frac{2}{3} - \frac{3}{7}$

STEP 5

Next, we need to subtract the fractions on the right side of the equation. The least common multiple of3 and7 is21, so we multiply the numerator and denominator of $\frac{2}{3}$ by7 and the numerator and denominator of $\frac{3}{7}$ by3.
$\frac{5}{42} x = \frac{14}{21} - \frac{9}{21}$

STEP 6

Subtract the fractions on the right side of the equation.
$\frac{5}{42} x = \frac{5}{21}$

STEP 7

Finally, to solve for $x$ , we divide both sides of the equation by $\frac{5}{42}$ .
$x = \frac{\frac{5}{21}}{\frac{5}{42}}$

STEP 8

implify the right side of the equation by multiplying the numerator and the reciprocal of the denominator.
$x = \frac{5}{21} \times \frac{42}{5}$

STEP 9

implify the right side of the equation to find the value of $x$ .
$x =2$ So, the solution to the equation $\frac{}{2} x+\frac{3}{7}=\frac{2}{3}+\frac{8}{21} x$ is $x =2$ .

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