Question

Simplify the expression: \((7 x-5 y+2 z)-(2 x-y+3 z)\) (A) \(5 x-6 y+5 z\) (B) \(5 x-4 y+5 z\) (C) \(5 x-4 y-z\) (D) \(9 x-4 y-z\)

Short answer

\(5 x-4 y-z\)

Step-by-step solution

Identify like terms

In the expression \((7x - 5y + 2z) - (2x - y + 3z)\), we can see that there are terms involving x, y, and z. We will perform addition/subtraction on those terms.

Distribute the negative sign

To simplify the expression, we need to distribute the negative sign to the terms inside the second parenthesis, which results in: \(7x - 5y + 2z - 2x + y - 3z\)

Combine like terms

Now, we can simplify the expression by combining the like terms: \((7x - 2x) + (-5y + y) + (2z - 3z)\)

Perform the operations

Perform the operations on each variable: \(5x - 4y - z\) The simplified expression is \(5x - 4y - z\). Comparing with the given options, we find the correct choice is: (C) \(5 x-4 y-z\)

Key concepts

Like Terms

In algebra, the concept of "like terms" is crucial when simplifying expressions. Like terms are terms that have the same variable raised to the same power. This means you can only combine terms that contain the exact same variables.
Understanding like terms makes it easier to simplify expressions because it shows us which parts of the expression can be combined. For instance, in the expression

• \(7x\) and \(-2x\) are like terms because they both have the variable \(x\).
• \(-5y\) and \(y\) are like terms because they each have the variable \(y\).
• \(2z\) and \(-3z\) are like terms because they contain the variable \(z\).

Combining like terms is all about adjusting the coefficients—the numbers in front of the variables—by performing addition or subtraction. Recognizing and organizing like terms can simplify complex expressions and help you solve equations more efficiently.

Distributive Property

The distributive property in algebra refers to the way we multiply a single term by terms within a parenthesis. It states that

• \(a(b + c) = ab + ac\).

This means you "distribute" the multiplication to each term inside the bracket.
In the given exercise, we use the distributive property by distributing a negative sign across the terms inside parentheses:

• The original expression \((7x - 5y + 2z) - (2x - y + 3z)\) involves distributing the negative sign to get:
• \(7x - 5y + 2z - 2x + y - 3z\).

By distributing the negative sign, we ensure that we transform the subtraction into the addition of opposite terms. This step is essential for correctly combining like terms and moving forward with simplifying the expression.

Simplifying Expressions

Simplifying an expression means making it as compact and clear as possible. This involves combining like terms and ensuring all operations are appropriately executed. Simplification is a fundamental arithmetic operation that prepares expressions for solving equations or modeling situations.
In our example, once we have distributed the negative sign and identified the like terms, the expression \(7x - 5y + 2z - 2x + y - 3z\) becomes a series of operations:

• Combine like terms for \(x\): \(7x - 2x = 5x\).
• Combine like terms for \(y\): \(-5y + y = -4y\).
• Combine like terms for \(z\): \(2z - 3z = -z\).

After combining these terms, the simplified expression is \(5x - 4y - z\). Simplification helps by removing any unnecessary complexity in expressions, making mathematical operations smoother and more straightforward for further calculations.