Question

Remove parentheses and simplify. $$(2 z+5 c-3 a)-(6 a+2 c-4 z)$$

Short answer

The simplified expression is \(6z + 3c - 9a\).

Step-by-step solution

Distribute the Negative Sign

Apply the distributive property of multiplication over subtraction to remove the parentheses around the second group of terms, changing their signs accordingly.

Combine Like Terms

After distribution, combine like terms by adding or subtracting the coefficients of the variables that match.

Simplify the Expression

Write the simplified expression with the combined like terms.

Key concepts

Distributive Property

When it comes to simplifying algebraic expressions, the distributive property is one of the fundamental tools you'll use. In essence, this property allows you to multiply a single term by each term inside a set of parentheses. The standard formula is \(a(b + c) = ab + ac\).

In our exercise, we face a slightly different scenario. We have to distribute a negative sign, which is the same as multiplying by -1. So when we have \( - (6a + 2c - 4z) \), applying the distributive property, we get \( -6a - 2c + 4z \). It's crucial to change the sign of each term inside the parentheses to maintain the expression's integrity.

• Positive becomes negative
• Negative becomes positive

This step sets up the equation for further simplification by ensuring all terms are correctly presented for combining.

Combining Like Terms

Once the distributive property has been applied, the next step is combining like terms. This process is quite straightforward: you look for terms that have the exact same variable to the same power and combine their coefficients. For example, terms with \(z\) should be combined with other \(z\) terms.

In our example, after distribution, we are left with the terms \(2z, 5c, -3a, -6a, -2c, \) and \(4z\). To combine like terms, just add or subtract the coefficients:

Combining \(a\) Terms

\( -3a - 6a = -9a \)

Combining \(c\) Terms

\(5c - 2c = 3c\)

Combining \(z\) Terms

\(2z + 4z = 6z\)
Remember, the variables don't change; only their coefficients do. This step simplifies the expression to a more manageable form and is necessary before any further algebraic manipulations.

Algebraic Manipulation

Algebraic manipulation is the overarching process that involves applying various rules and properties of algebra to manipulate and simplify expressions. It's an art form in the math world. In our case, after applying the distributive property and combining like terms, we arrive at the simplified form of the expression.

Let's look back at our example. After distributing the negative sign and combining like terms, we get the simplified expression \(6z + 3c - 9a\). The expression is now easier to understand and use in further calculations or equations.

It's good practice to look for opportunities to factor or expand expressions or even to cancel out terms across an equation. Each of these skills falls under the broad umbrella of algebraic manipulation and is essential for mastering algebra and advancing in mathematical understanding.