Question $\frac{1}{3} x+1=-x+\frac{1}{5}$

Studdy Solution

STEP 1

1. The equation $\frac{1}{3} x + 1 = -x + \frac{1}{5}$ is a linear equation with one variable.
2. We need to find the value of $x$ that satisfies the equation.
3. The equation involves fractions and basic algebraic operations.

STEP 2

1. Eliminate fractions by finding a common denominator.
2. Combine like terms to simplify the equation.
3. Isolate the variable $x$ to solve the equation.
4. Check the solution by substituting it back into the original equation.

STEP 3

To eliminate the fractions, find the least common denominator (LCD) of the fractions involved. The denominators are 3 and 5, so the LCD is 15.
Multiply every term in the equation by 15 to clear the fractions:
$15 \left(\frac{1}{3} x\right) + 15(1) = 15(-x) + 15\left(\frac{1}{5}\right)$

STEP 4

Simplify each term:
$5x + 15 = -15x + 3$

STEP 5

Combine like terms by moving all terms involving $x$ to one side and constant terms to the other side. Add $15x$ to both sides:
$5x + 15x + 15 = 3$

STEP 6

Simplify the equation:
$20x + 15 = 3$

STEP 7

Isolate $x$ by subtracting 15 from both sides:
$20x = 3 - 15$

STEP 8

Simplify the right-hand side:
$20x = -12$
Divide both sides by 20 to solve for $x$ :
$x = \frac{-12}{20}$

STEP 9

Simplify the fraction:
$x = \frac{-3}{5}$

STEP 10

Check the solution by substituting $x = \frac{-3}{5}$ back into the original equation:
$\frac{1}{3}\left(\frac{-3}{5}\right) + 1 = -\left(\frac{-3}{5}\right) + \frac{1}{5}$

STEP 11

Simplify both sides:
Left side:
$\frac{-1}{5} + 1 = \frac{-1}{5} + \frac{5}{5} = \frac{4}{5}$
Right side:
$\frac{3}{5} + \frac{1}{5} = \frac{4}{5}$
Since both sides are equal, the solution is verified.
The solution is:
$\boxed{\frac{-3}{5}}$

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