Problem 44 - Prealgebra

Question

The following equations contain parentheses. Apply the distributive property to remove the parentheses, then simplify each side before using the addition property of equality. $$4(2 a-1)-7 a=9-5$$

Step-by-Step Solution

Answer

The solution is \(a = 8\).

Step 1: Apply the Distributive Property

To start, apply the distributive property to the expression on the left-hand side. The equation is \( 4(2a - 1) - 7a = 9 - 5 \). Distribute the 4 across \( 2a - 1 \) to get \( 8a - 4 \). Thus, the equation becomes: \( 8a - 4 - 7a = 9 - 5 \).

Step 2: Simplify Both Sides

Next, simplify both sides of the equation. On the left-hand side, combine like terms \(8a - 7a\) to get \(a\), and simplify \(-4\) to keep it the same. Thus, the left-hand side becomes \(a - 4\). On the right-hand side, calculate \(9 - 5\), which equals 4. The equation now is \(a - 4 = 4\).

Step 3: Use the Addition Property of Equality

Use the addition property of equality to isolate the variable \(a\). Since the current equation is \(a - 4 = 4\), add 4 to both sides to eliminate the \(-4\) on the left side. This results in \(a = 8\).

Key Concepts

Addition Property of Equality

Imagine you're holding a balance scale. Just like in a math equation, you can freely add or subtract the same amount on both sides, and the balance stays perfectly even. This idea is the core of the Addition Property of Equality.
In math terms, it means if you have an equation like \(x - 3 = 5\), you can add 3 to both sides. This operation doesn't change the equation's balance—it simply helps in isolating the variable.

• For example: add 3 to both sides of \(x - 3 = 5\).
• This yields: \(x - 3 + 3 = 5 + 3\).
• Simplifying gives \(x = 8\).

Always remember: whatever you do to one side, do to the other. This consistency ensures you solve equations accurately by maintaining equality.

Combining Like Terms

Combining like terms is like organizing a messy drawer. It helps simplify long expressions to make them easier to handle.
Like terms are terms that have the same variable raised to the same power. For instance, in the term \(8a - 7a\), both parts contain the variable 'a'. This commonality allows you to combine them.

• Combine them by simply operating on the coefficients: \(8a - 7a\) becomes \(1a\) or just \(a\). This step reduces complexity without changing the equation's value.
• The process is similar with constants. Simply add or subtract the constant numbers: In \(-4 + 0\), the result is \(-4\).

By combining these terms, you simplify your equation, which is a crucial part of solving it.

Solving Linear Equations

Solving linear equations is like unraveling a mystery where you need to find the value of an unknown. Linear equations, like \(a - 4 = 4\), involve only one variable and no exponents.
Think of solving an equation as peeling away layers until you reach the core, the solution for the variable.

• First, apply simple operations like addition or subtraction to isolate the variable. In \(a - 4 = 4\), adding 4 to both sides isolates \(a\).
• Break down complex parts using properties like the Distributive Property if needed, as seen when simplifying equations like \(4(2a - 1)\).
• Once simplified, use logical steps to clear surrounding numbers or terms, leaving the variable isolated.

The beauty of linear equations is their straightforward nature. Master these principles, and you can easily solve similar problems.