Problem 96 - Intermediate Algebra

Question

Solve each equation. $$ 2[5(4-a)+2(a-1)]=3-a $$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 6.6\).

Step 1: Distribute within the brackets

First, handle the operation inside the brackets. Distribute the numbers inside the brackets: 1. For the first part inside the brackets: \[5(4-a) = 20 - 5a\] 2. For the second part inside the brackets: \[2(a-1) = 2a - 2\]Combine these results inside the brackets: \[20 - 5a + 2a - 2\] simplifies to \[18 - 3a\] after combining like terms.

Step 2: Apply the factor outside the brackets

Take the simplified expression inside the brackets, \[18 - 3a\], and distribute the 2 outside of the brackets:\[2(18 - 3a) = 36 - 6a\]

Step 3: Set the equation to equal the right side

Set the distributed equation from Step 2 equal to the right side of the equation:\[36 - 6a = 3 - a\]

Step 4: Simplify the equation

Move all terms involving \(a\) to one side and constant terms to the other side:Add \(6a\) and subtract 3 from both sides:\[36 - 3 = 6a - a\]Which simplifies to:\[33 = 5a\]

Step 5: Solve for a

Solve for \(a\) by dividing both sides by 5:\[a = \frac{33}{5}\]Therefore, \(a = 6.6\).

Key Concepts

Distribution in Algebra

Distribution in algebra involves the multiplication of a single term across terms within parentheses. It is crucial for transforming and simplifying algebraic expressions.

Let's imagine an equation where a number is outside of parentheses, like this: \( 2(3x + 4) \). The idea is to multiply the number outside the parentheses by each term inside. So, it would result in:

• \( 2 \times 3x = 6x \)
• \( 2 \times 4 = 8 \)

Putting these results together, the expression becomes \( 6x + 8 \). This process ensures each term inside the parentheses is fully considered and multiplied by the external term, which is the essence of distribution.

It's helpful because it allows us to eliminate the parentheses and move towards solving the equation. Whenever you see parentheses in an algebraic problem, think about distribution as a key strategy to simplify your work.

Combining Like Terms

Combining like terms helps simplify expressions by adding or subtracting terms that have the same variable part. For instance, in the expression \(4x - 2x + 3\), "like terms" are the parts that involve the same variable (in this case, \(x\)).

To combine like terms:

• Identify terms with the same variable(s).
• Add or subtract the coefficients of these terms.

In our example, \(4x - 2x\) are like terms. You simply subtract their coefficients to get \(2x\). That makes the expression \(2x + 3\).

This technique simplifies complex expressions into more manageable forms, making it easier to isolate and solve for variables. By reducing the number of terms, you'll have a clearer pathway to the solution.

Simplifying Equations

Simplifying equations is a core algebraic technique to make equations easier to solve. It involves a combination of distribution, combining like terms, and isolating variables.

Here's a general strategy:

• First, use distribution to eliminate parentheses if necessary.
• Next, combine like terms on each side of the equation.
• Then, get all terms involving the variable on one side and constants on the other.

For example, consider the equation \(36 - 6a = 3 - a\). You aim to simplify by rearranging terms.

• Add \(6a\) to both sides to get \(36 = 3 + 5a\).
• Then, subtract 3 from both sides to isolate terms with \(a\): \(33 = 5a\).
• Finally, simplify further by dividing by \(5\) to solve for \(a\): \(a = 6.6\).

This step-by-step simplification reduces complexities and leads you directly to the answer. Always remember, the goal is to make the equation as simple as possible, focusing on isolating the variable to solve it efficiently.