Problem 57 - Algebra A Combined Function

Question

Solve each equation. See Section \(2.3 .\) \(8 a-2(3 a-1)=6\)

Step-by-Step Solution

Answer

The solution is \(a = 2\).

Step 1: Distribute the Negative Sign

First, apply the distributive property to the expression on the left side. The equation is currently \(8a - 2(3a - 1) = 6\). Distribute the \(-2\) over the terms inside the parentheses: \(-2 \times 3a = -6a\) and \(-2 \times -1 = 2\). This changes the equation to \(8a - 6a + 2 = 6\).

Step 2: Simplify the Equation

Combine like terms in the equation's left side. You have \(8a - 6a\), which simplifies to \(2a\). So, the equation becomes \(2a + 2 = 6\).

Step 3: Isolate the Variable Term

Subtract 2 from both sides of the equation to isolate the variable term. This gives you \(2a = 4\).

Step 4: Solve for the Variable

Divide both sides of the equation by 2 to solve for \(a\). So, \(a = 4/2 = 2\).

Key Concepts

Distributive Property

The distributive property is a useful tool in algebra that helps to simplify equations. In the given problem, we start with the equation:
\(8a - 2(3a - 1) = 6\).In this instance, the distributive property allows us to multiply a single term across terms within parentheses. In this equation,
-2 is distributed over both \(3a\) and \(-1\). Here's what happens during distribution:

• When \(-2\) is multiplied by \(3a\), it gives \(-6a\).
• Then, multiplying \(-2\) by \(-1\) results in \(+2\).

After employing the distributive property, our equation is transformed into:
\(8a - 6a + 2 = 6\). This simplification through distribution reduces the complexity of the equation and prepares us for the next steps.

Combining Like Terms

Combining like terms is a strategy used to simplify equations by merging terms with the same variable and exponent. After applying the distributive property, the equation becomes:
\(8a - 6a + 2 = 6\). Both \(8a\) and \(-6a\) share the variable \(a\). When we combine them:

• The \(8a\) and \(-6a\) add up to \(2a\).
• The remaining constant "+2" stays unchanged.

This simplifies the equation to:
\(2a + 2 = 6\). By combining like terms,
we make the algebraic equation shorter and easier to solve,
allowing us to focus on solving for the variable next.

Isolating Variables

Isolating the variable is the process of solving the equation by "getting the variable alone" on one side of the equation. After simplifying the previous equation, we have:
\(2a + 2 = 6\). To isolate \(a\), we must remove any constants or coefficients alongside it:

• First, subtract "2" from both sides of the equation.
  This gives us \(2a = 4\).
• Next, divide both sides by \(2\) to finally isolate \(a\).

Resulting in \(a = 2\). Isolating the variable is crucial because it allows us to accurately find the value of the unknown in the equation,
providing clarity and solution to the original problem. As you practice,
this step will become second nature.