Math Snap

STEP 1

1. The equation $\frac{1}{5}x + 3 = -\frac{1}{4}x + 3$ is a linear equation in one variable.
2. The goal is to solve for $x$.
3. The equation involves fractions, which require careful manipulation.

STEP 2

1. Simplify the equation by eliminating the constant terms on both sides.
2. Combine like terms to isolate the variable $x$.
3. Solve for $x$.

STEP 3

First, observe that both sides of the equation have a constant term $+3$. We can subtract $3$ from both sides to simplify the equation:
$$\frac{1}{5}x + 3 - 3 = -\frac{1}{4}x + 3 - 3$$ This simplifies to:
$$\frac{1}{5}x = -\frac{1}{4}x$$

STEP 4

Next, we need to combine the $x$-terms. To do this, add $\frac{1}{4}x$ to both sides to get all $x$-terms on one side:
$$\frac{1}{5}x + \frac{1}{4}x = 0$$

STEP 5

To combine the fractions, find a common denominator. The common denominator of 5 and 4 is 20. Convert each fraction:
$$\frac{4}{20}x + \frac{5}{20}x = 0$$ Combine the fractions:
$$\frac{9}{20}x = 0$$

To solve for $x$, multiply both sides by the reciprocal of $\frac{9}{20}$, which is $\frac{20}{9}$:
$$x = 0 \times \frac{20}{9}$$ Simplify:
$$x = 0$$ The solution is:
$$\boxed{0}$$