Problem 31 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3(2 x-1)-4(x+2)-5(3 x+4) $$

Step-by-Step Solution

Answer

The simplified expression is \(-13x - 31\).

Step 1: Distribute Multiplication

Apply the distributive property to remove the parentheses by multiplying each term inside by the factor outside: Start with \(3(2x - 1)\), which becomes \(6x - 3\).Next, for \(-4(x + 2)\), multiply to get \(-4x - 8\).Finally, for \(-5(3x + 4)\), distribute to get \(-15x - 20\).The expression is now: \(6x - 3 - 4x - 8 - 15x - 20\).

Step 2: Combine Like Terms

Combine the like terms of the expression:The terms involving \(x\) are \(6x, -4x,\) and \(-15x\). Combine them:\(6x - 4x - 15x = -13x\).Now, combine the constant terms: \(-3 - 8 - 20 = -31\).Thus, the simplified expression becomes \(-13x - 31\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps us simplify expressions by removing parentheses. It states that if you have an expression like \(a(b + c)\), you can multiply \(a\) with both \(b\) and \(c\). This results in \(ab + ac\).

Here's how we applied the distributive property in our example:

• For the expression \(3(2x - 1)\), we multiplied \(3\) by both \(2x\) and \(-1\). This gave us \(6x - 3\).
• Next, \(-4(x + 2)\), the \(-4\) multiplies with \(x\) and \(2\), resulting in \(-4x - 8\).
• Last, \(-5(3x + 4)\) becomes \(-15x - 20\) when \(-5\) is distributed to both terms inside the parenthesis.

Using this property makes expressions easier to handle and sets the stage for combining like terms.

Combining Like Terms

Once parentheses are removed, our next step is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. This means you can add or subtract their coefficients.

In the expression, for terms involving \(x\), we had \(6x\), \(-4x\), and \(-15x\). They are all like terms because they contain the same variable \(x\). By combining their coefficients, we get:

• \(6x - 4x - 15x = -13x\)

Next, for the constant terms that don't have variables, \(-3\), \(-8\), and \(-20\), we combine them as follows:

• \(-3 - 8 - 20 = -31\)

Combining like terms simplifies the expression to something easier to manage. This process is crucial for making complex expressions more manageable.

Simplification

Simplification in algebra involves turning complex expressions into their simplest form. By using the distributive property and combining like terms, we arrive at a refined expression that is easier to understand and use in further calculations.

After applying the distributive property and combining like terms in our example, the expression \(3(2x-1)-4(x+2)-5(3x+4)\) has been reduced to \(-13x-31\).

This does not only make the calculations clearer but also ensures that you can spot errors more easily and interpret the expression readily.

• It allows us to see the net effect of all operations being performed.
• Simplification transforms complex algebraic expressions into basic forms that are much more manageable and easy to work with, whether in solving equations or modeling real-world situations.

Simplified expressions are less prone to mistakes, especially useful when working through more challenging algebraic problems.