Problem 22 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 8(x+5)=7(x+8) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 16.\)

Step 1: Distribute

Multiply all terms inside the parentheses by the number outside. This results into \(8x+ 40 = 7x + 56.\)

Step 2: Consolidate Variables

Move all terms involving the variable \(x\) to one side of the equation by subtracting \(7x\) from both sides. Accordingly, this results into \(x + 40 = 56.\)

Step 3: Solve for the Variable

Solve for \(x\) by subtracting 40 from both sides of the equation to isolate the variable on one side. This results into \(x = 16.\)

Key Concepts

Distributive Property

The distributive property is a key principle in algebra which allows us to multiply a single term across a sum or difference within parentheses. Consider an expression like 8(x+5). To apply the distributive property, you multiply 8 by each term inside the parentheses individually: 8 times x and 8 times 5. This gives us 8x + 40.

Understanding and applying this property is crucial when solving algebraic equations because it simplifies complex expressions and prepares the equation for further solving steps. It's especially useful in eliminating the parentheses to allow the consolidation of like terms and to progress toward isolating the variable.

Isolating Variables

Isolating the variable means manipulating an equation in such a way that the variable you're solving for is by itself on one side of the equation. In simple terms, you want the 'x' or any other variable to be alone.

For instance, after distributing and simplifying, we have an equation like x + 40 = 56. To get x by itself, we subtract 40 from both sides of the equation. This gives us x = 56 - 40, which simplifies to x = 16. It's like solving a puzzle where you methodically move pieces (in this case, numbers or terms) until the picture (solution) is clear.

Equation Balancing

Balancing an equation is crucial to solving for the unknown and ensuring the equality remains true. It involves doing the same mathematical operation to both sides of the equation.

Think of the equation as a scale with balances; whatever you do to one side, you must do to the other to keep it balanced. In our exercise, to consolidate variables, we subtract 7x from each side to eliminate x's from the right side. This maneuver is essential for maintaining the balance of the equation while moving towards the solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In our example, 8x+40 and 7x+56 are both algebraic expressions. Understanding these expressions and how to manipulate them through operations like addition, subtraction, multiplication, and division is foundational.

Getting comfortable with transforming and interpreting these expressions is fundamental, as they are the building blocks upon which equations are formed and solved. Recognizing patterns and the structure of these expressions can greatly aid in mastering algebraic problem-solving.