Question

Add: \(a-c+b\) and \(b+c-a\)

Short answer

The short answer is: \(2b\)

Step-by-step solution

Write down the expressions to be added

List both expressions that need to be added: Expression 1: \(a-c+b\) Expression 2: \(b+c-a\)

Group like terms together

Rearrange the terms of each expression so that like terms are together and it becomes easier to combine them. Expression 1: \(a + b - c\) Expression 2: \(b + c - a\)

Combine like terms

Add the corresponding like terms together from both expressions. Combine terms with \(a\), with \(b\), and with \(c\) separately. \( (a - a) + (b + b) + (-c + c) \)

Key concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra. They are combinations of variables, numbers and operations that represent a specific quantity. Consider the expression \(2x + 3y - 5\). Here, \(x\) and \(y\) are the variables representing quantities that can change, while the numbers 2, 3, and -5 are constants that give us information about how much the variables contribute to the value of the expression.

An essential skill when working with algebraic expressions is the ability to manipulate and simplify them. This means combining like terms, factoring, expanding, and applying various algebraic properties to produce an equivalent expression that's usually simpler to work with or solve.

Addition of Polynomials

Polynomials are algebraic expressions that include terms with variables raised to whole-number exponents. The addition of polynomials entails combining similar terms from two or more polynomials to create a single polynomial. For example, adding \(3x^2 + 2x + 1\) and \(4x^2 - 3x + 5\) involves combining the \(x^2\) terms, the \(x\) terms, and the constant terms. To do this efficiently, we arrange terms with the same variables and exponents close to each other, simplifying our work. The result in this case would be \(7x^2 - x + 6\), a simpler, single polynomial.

Understanding the addition of polynomials is key for many areas of mathematics, including solving equations and understanding functions.

Grouping Like Terms

Grouping like terms is a method used to simplify algebraic expressions. It involves rearranging the terms so that those with the same variable part are placed together. The aim is to make the addition or subtraction of terms more straightforward. For instance, in the expressions \(a - c + b\) and \(b + c - a\), we rearrange the terms: \((a + b - c\)) and \((b + c - a)\) respectively.

The 'like terms' here are those that have the exact same variables to the same power. In our example, terms with \(a\), terms with \(b\), and terms with \(c\) are grouped, then added or subtracted accordingly. This method forms the basis of simplifying expressions and solving algebraic equations.