Simplify (x^2+y^2)+(-x^2+y^2)
The problem is asking to perform algebraic simplification on the given expression by combining like terms. The expression includes two binomials contained within parentheses: the first is \(x^2 + y^2\) and the second is \(-x^2 + y^2\). The operation to be carried out is the addition of these two binomials, and during the simplification process, it is necessary to combine terms that are alike (terms that have the same variable raised to the same power). This operation is expected to simplify the expression to an easier or more reduced form.
$\left(\right. x^{2} + y^{2} \left.\right) + \left(\right. - x^{2} + y^{2} \left.\right)$
Answer
Solution:
Step:1
Eliminate the parentheses to get $x^{2} + y^{2} - x^{2} + y^{2}$.
Step:2
Identify and combine like terms in the expression $x^{2} + y^{2} - x^{2} + y^{2}$.
Step:2.1
Cancel out $x^{2}$ with $-x^{2}$ to obtain $0 + y^{2} + y^{2}$.
Step:2.2
Combine $y^{2}$ with $0$ to maintain the expression as $y^{2} + y^{2}$.
Step:3
Finally, add together the two $y^{2}$ terms to get $2y^{2}$.
Knowledge Notes:
To simplify an algebraic expression, you follow a series of steps:
Eliminate Parentheses: Apply the distributive property, if necessary, to remove any parentheses.
Combine Like Terms: Group together and simplify terms that contain the same variable raised to the same power. In algebraic expressions, like terms are terms that have the same variables and powers. The coefficients of these terms can be added or subtracted from one another.
Simplify the Expression: After combining like terms, the expression should be simplified to its most reduced form.
In the given problem, we are simplifying the expression $(x^2+y^2)+(-x^2+y^2)$. The process involves removing the parentheses, which is straightforward since there are no coefficients other than 1 affecting the terms inside the parentheses. Next, we combine like terms by subtracting $x^2$ from $x^2$, which cancels out the $x^2$ terms, leaving only the $y^2$ terms. We then add the $y^2$ terms together to get the final simplified result of $2y^2$.
The use of LaTeX in the solution is to clearly represent mathematical expressions and ensure that the variables, exponents, and coefficients are properly formatted for clarity and ease of understanding.
