Question

If $x\ne 0$ and $x-\frac{2-x^2}{x}=\frac{y}{x},$ then $y=$

Short answer

y = 2x^2 - 2

Step-by-step solution

- Simplify the Left Side of the Equation

Start by simplifying the left side of the equation: $$x-\frac{2-x^2}{x}$$ First, rewrite the fraction: $$x-\frac{2}{x}+\frac{x^2}{x}$$ This simplifies to: $$x-\frac{2}{x}+x$$

- Combine Like Terms

Combine the like terms in the expression:$$x-\frac{2}{x}+x=2x-\frac{2}{x}$$

- Set the Simplified Left Side Equal To the Right Side

Now set the simplified left side of the equation equal to the right side given in the problem: $$2x-\frac{2}{x}=\frac{y}{x}$$

- Eliminate the Denominator

Multiply both sides of the equation by $x$ to eliminate the denominator: $$x(2x-\frac{2}{x})=y$$ This simplifies to: $$2x^2-2=y$$

- Solve for y

The equation is now simplified to: $$y=2x^2-2$$

Key concepts

simplifying algebraic expressions

Simplifying algebraic expressions is the process of making an expression easier to work with by combining and reducing terms. In our given problem, the expression is simplified to help solve for the variable using algebraic rules.
Let's see this with an example:
The original expression is: $$x-\frac{2-x^2}{x}$$First, let's rewrite the fraction: $$x-\frac{2}{x}+\frac{x^2}{x}$$ Here, we separate the numerator terms and distribute the denominator across them, which gives us: $$x-\frac{2}{x}+x$$
Simplification involves breaking an expression into simpler parts before combining like terms.
This results in a more straightforward expression that is easier to handle.

combining like terms

Combining like terms is a fundamental skill in algebra critical for simplifying expressions. Let's see this with the simplified expression from our previous step: $$x-\frac{2}{x}+x$$ Here, the terms containing 'x' should be combined:
1. Identify the 'like terms'—terms that have the same variable raised to the same power. 2. Combine them by adding or subtracting their coefficients.
Combining $x\text{ and }x\to 2x$This leaves us with: $$2x-\frac{2}{x}$$
By combining like terms, the expression is more concise and manageable.

eliminating denominators

Eliminating denominators simplifies equations by removing fractions, making it easier to solve for a variable.
In our example, the simplified equation is:
$$2x-\frac{2}{x}=\frac{y}{x}$$
To eliminate the denominator, multiply every term by the common denominator, in this case, 'x':
$$x(2x-\frac{2}{x})=y$$ This step simplifies the equation by removing the fractions: $$2x^2-2=y$$ Now the equation is reduced to an algebraic expression without fractions, making it easier to solve for y: $y=2x^2-2$ Removing fractions is crucial for simplifying and solving algebraic equations efficiently.