Problem 80 - Algebra 2

Question

Solve each equation. $$ 2(a-6)+11=25 $$

Step-by-Step Solution

Answer

\(a = 13\)

Step 1: Expand the Parentheses

To start solving the equation, expand the expression on the left-hand side by applying the distributive property. Multiply 2 by each term inside the parentheses: \(2(a-6) = 2 \times a + 2 \times (-6) = 2a - 12\).

Step 2: Simplify the Equation

Next, add the result from expanding the parentheses to the 11 on the left-hand side to simplify the equation: \(2a - 12 + 11 = 25\). This simplifies to \(2a - 1 = 25\).

Step 3: Add 1 to Both Sides

To isolate the term containing \(a\) on one side, add 1 to both sides of the equation: \(2a - 1 + 1 = 25 + 1\), which simplifies to \(2a = 26\).

Step 4: Divide Both Sides by 2

Lastly, to solve for \(a\), divide both sides of the equation by 2: \(2a / 2 = 26 / 2\), which simplifies to \(a = 13\).

Key Concepts

Distributive Property

Understanding the distributive property is crucial for simplifying and solving algebraic equations. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For instance, in the equation \(2(a-6)+11=25\), the distributive property is applied to expand \(2 \times (a - 6)\). This means you distribute the multiplication of 2 across both \(a\) and \(6\), leading to the equation \(2a - 12\).

This property is your first step to breaking down and simplifying complex algebraic expressions, ensuring that every term is considered before you proceed to further simplification and solving.

Equation Simplification

After applying the distributive property, the next key concept is equation simplification. This involves combining like terms and reducing the equation to its simplest form. In our example, after expanding the parentheses, we combine the constant terms (\( -12 + 11\)) on the left side, and the equation simplifies to \(2a - 1 = 25\).

By simplifying the equation, it becomes more manageable and sets the stage for clearer steps towards finding the solution. Simplification can also reveal potential errors early in the problem-solving process, allowing for corrections before further calculations.

Isolating Variables

To solve for a variable means to isolate it on one side of the equation. In the exercise, to isolate \(a\), we progress from the simplified form \(2a - 1 = 25\) and perform operations that 'undo' the other terms affecting \(a\). Firstly, adding 1 to both sides negates the \( -1\) and results in \(2a = 26\).

It's crucial to maintain the balance of the equation by performing operations equally on both sides. This way, the equality is preserved, and we step closer to finding the value of the isolated variable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They represent values that can vary depending on the variables involved. In \(2(a - 6) + 11 = 25\), \(2(a - 6) + 11\) is an algebraic expression that becomes simpler as we apply the distributive property and combine like terms.

Manipulating these expressions correctly is essential to solving equations. The aim is to translate the expression into a form where the variable's value becomes clear. By understanding and manipulating algebraic expressions, we transform complex problems into simple, solvable equations.