91Ó°ÊÓ

Algebraic equations involve variables, coefficients, and constants combined using mathematical operations. In our example, the equation is \( \frac{1}{3} x + \frac{1}{2} x = \frac{1}{6} x \). Here, x is a variable, and the fractions are coefficients. The goal is to solve for the variable. Understanding how to manipulate these elements through operations like addition, subtraction, multiplication, and division is crucial. Algebraic equations can be linear, polynomial, or more complex. For beginners, it's helpful to start with linear equations like the one we are working with to build a strong foundation.

Fractional coefficients are fractions used as multipliers for the variable in an equation. In our problem, the coefficients are \( \frac{1}{3} \), \( \frac{1}{2} \), and \( \frac{1}{6} \). These can complicate the equation, but multiplying by the least common denominator (LCD) or another strategic number can clear them.

For example, multiplying the entire equation by 12 clears all the fractions as follows: \[ 12 \cdot \frac{1}{3} x + 12 \cdot \frac{1}{2} x = 12 \cdot \frac{1}{6} x \].

Each term on the left simplifies to 4x and 6x while the right simplifies to 2x. This transformation makes the equation easier to solve, reducing the complexity caused by the fractional coefficients.

Multiplication is a powerful tool in solving equations, especially when dealing with fractions. By multiplying every term in the equation by a common number, you can simplify the fractions and work with whole numbers instead.

In our equation \( \frac{1}{3} x + \frac{1}{2} x = \frac{1}{6} x \), we decided to multiply by 12. This means:

• \ 12 \cdot\ \frac{1}{3} x = 4x
• \ 12 \cdot\ \frac{1}{2} x = 6x
• \ 12 \cdot\ \frac{1}{6} x = 2x

Using multiplication helps to eliminate the fractions, making it simpler to collect like terms and balance both sides of the equation. Understanding when and how to apply multiplication is essential for simplifying and solving algebraic equations.

Simplifying an equation involves combining like terms and performing operations to isolate the variable. The ultimate goal is to make the equation solvable with the variable on one side and constants on the other.

Initially, we had \( \frac{1}{3} x + \frac{1}{2} x = \frac{1}{6} x \). After multiplying through by 12, we simplified to:

• Left side: 4x + 6x = 10x
• Right side: 2x

This left us with \( 10x = 2x \). Subtracting 2x from both sides results in 8x = 0. Dividing by 8 simplifies it further to x = 0.

Equation simplification makes solving algebraic equations clear and concise. You get to see each step leading to the final solution. This process highlights the importance of organized, step-by-step operations to reach the correct solution.