Problem 2 - Elementary Algebra

Question

Solve each equation. $$4(x+4)=24$$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

The given equation is \(4(x+4) = 24\). This is a linear equation where we need to find the value of \(x\) that satisfies the equation.

Step 2: Apply the Distributive Property

Apply the distributive property by multiplying 4 with each term inside the parentheses: \(4 \cdot x + 4 \cdot 4 = 24\). This simplifies to \(4x + 16 = 24\).

Step 3: Isolate the Variable Term

To isolate the variable term, subtract 16 from both sides of the equation: \(4x + 16 - 16 = 24 - 16\). This simplifies to \(4x = 8\).

Step 4: Solve for x

Divide both sides by 4 to solve for \(x\): \(\frac{4x}{4} = \frac{8}{4}\). This simplifies to \(x = 2\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, we use letters like \(x\) to stand in for values that can change. These letters are known as variables. Equations like \(4(x + 4) = 24\) are called linear equations because their graph forms a straight line. When solving algebraic problems, the goal is to find the values for the variables that make the equation true. Algebra helps us to understand and describe patterns or relationships. It is a foundational element for higher-level math and real-world problem-solving. You can think of algebra as a puzzle where you use logical steps to find the missing piece.

Distributive Property

The distributive property is a useful mathematical property used to simplify expressions. It states that a number multiplied by a sum of two or more numbers is equal to the sum of individual products of the number and each addend in the parentheses. For example, the expression \(a(b + c)\) can be rewritten as \(ab + ac\). In our equation \(4(x + 4) = 24\), applying the distributive property means multiplying 4 by each term inside the parentheses:

• First, multiply \(4\) by \(x\), which results in \(4x\).
• Next, multiply \(4\) by \(4\), which gives \(16\).

Thus, the equation becomes \(4x + 16 = 24\). Understanding and applying this property simplifies and opens the path to solve the equation for the unknown variable quickly.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Consider the simplified equation \(4x + 16 = 24\). Here, we need to isolate \(x\), the variable we wish to solve for.A good strategy is to perform inverse operations to achieve this:

• First, focus on the constant term by subtracting \(16\) from both sides. This gives us \(4x = 8\).
• Next, divide each side by \(4\) to solve for \(x\), resulting in \(x = 2\).

Each step ensures that the equation remains balanced just like a scale, maintaining equality. Solving equations is a critical skill in algebra, as it forms the basis for handling more complex mathematical problems and real-life situations where conditions are stated in terms of relationships between unknowns and known values.