Question

Solve the equation. $$ \frac{x}{-6}=-\frac{1}{2} $$

Short answer

The solution to the equation is \(x = -6\).

Step-by-step solution

Cross Multiply

Cross multiply the fractions. This is done by multiplying the denominator of the left fraction with the numerator of the right fraction, and vice versa. \(x \times -\frac{1}{2} = -6 \times -\frac{1}{2}\)

Simplify

Simplify the equation to isolate \(x\). \(-\frac{1}{2}x = 3\)

Solving for x

Multiply both sides of the equation by -2 to solve for \(x\). \(x = -6\)

Key concepts

Cross Multiplication

Cross multiplication is a vital technique to solve equations involving two fractions set equal to each other. The method simplifies the process of finding a common denominator, allowing students to easily clear the fractions.

Consider the equation \( \frac{x}{-6}=-\frac{1}{2} \). To use cross multiplication, we multiply the numerator of one fraction by the denominator of the other, and vice versa. The equation becomes \( x \times -\frac{1}{2} = -6 \times -\frac{1}{2} \).

It's critical to handle the signs carefully. Here, as both terms are negative, their product becomes positive. The cross multiplication transforms the original proportion into a much simpler equation without fractions, paving the way for isolating the variable.

Isolate the Variable

Isolating the variable is a principal goal when solving linear equations. It involves arranging the equation to have the variable on one side and all other elements on the opposite side. This practice allows for a clear understanding of the variable's value.

In the given example, we aim to isolate \(x\). After cross multiplication, the equation \( -\frac{1}{2}x = 3 \) is obtained. To isolate \(x\), we need to get rid of the \( -\frac{1}{2} \) that is multiplying the variable. This is done by doing the inverse operation; since \(x\) is multiplied by \( -\frac{1}{2} \), we do the opposite and multiply by \( -2 \) to get \( x \) alone. Isolating the variable is a crucial skill in algebra, as it neatly displays the solution to the equation.

Simplification in Algebra

Simplification is the process of tidying up an algebraic expression or equation, making it easier to work with. In the process of solving an equation, simplification often leads to an easier path to the solution by consolidating like terms and reducing complex expressions down to simpler forms.

Take for instance our equation after cross multiplication, \(x = -6 \times -\frac{1}{2}\). Simplifying the right-hand side involves multiplying the numbers, which results in \(x = 3\). In algebra, simplification can include combining like terms, reducing fractions, or clearing parentheses. It's the algebraic equivalent of cleaning your room, making sure everything is in the right place for a clearer perspective.