Problem 34 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(11 x+13=9 x+9\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2\).

Step 1: Simplify both sides

First, let's look at the equation: \(11x + 13 = 9x + 9\). The goal is to gather all \(x\)-terms on one side and constants on the other side.

Step 2: Move x-terms to one side

Subtract \(9x\) from both sides to get the \(x\)-terms on one side: \[11x + 13 - 9x = 9x + 9 - 9x\]This simplifies to: \[2x + 13 = 9\]

Step 3: Isolate the variable

Next, isolate \(2x\) by subtracting \(13\) from both sides:\[2x + 13 - 13 = 9 - 13\]This simplifies to:\[2x = -4\]

Step 4: Solve for x

Now, solve for \(x\) by dividing both sides by \(2\):\[x = \frac{-4}{2}\]This results in:\[x = -2\]

Key Concepts

Understanding Algebra

Algebra isn’t just about completing computations; it’s the language of mathematics, allowing us to describe patterns and solve problems involving numbers. The basic components in algebra include constants, variables, and operations. Variables, often represented by letters like \( x \), stand in for unknown values that we aim to determine. Constants are fixed values, such as numbers. Operations like addition, subtraction, multiplication, and division help us relate variables and constants to one another.

Algebraic equations are like puzzles to be solved. They might tell us that two expressions are equal. For example, in the exercise, the equation is \( 11x + 13 = 9x + 9 \), which states that the left side of the equation is equal to the right side. Understanding this relationship is crucial for solving any algebraic equation. We seek to find all possible values of \( x \) that make this equation true.

• Utilize the operations and rules of algebra to simplify and solve equations.
• Understand the function of each term in the equation for greater clarity.

Solving Equations

Solving equations is a methodical process. It involves working step by step to find the value of the variable that satisfies the equation. Let's take the equation \( 11x + 13 = 9x + 9 \). This step-by-step approach is the foundation of understanding how equations function.

• Simplification is key: Look at both sides of the equation and simplify if possible by combining like terms or performing arithmetic operations.
• Balance the equation: Whatever operation you do to one side, do the same to the other side to maintain equality. It's like a seesaw — balance is essential!

Starting with simplifying both sides and then getting all \( x \)-terms on one side, this exercise helps us systematically work through the problem until the value of \( x \) is isolated and identified.

Variable Isolation

Variable isolation is the art of making the equation simpler by getting the unknown variable on one side of the equation. Think of it like peeling away layers to reach the core of the problem. In the example \( 11x + 13 = 9x + 9 \), the first goal is to get all \( x \)-terms together.

We start by subtracting \( 9x \) from both sides to group all \( x \)-terms on one side:

• This transforms the equation to \( 2x + 13 = 9 \).

Next, move the constants to isolate \( x \). Subtract \( 13 \) from both sides to clear the number from the variable's side:

• This results in \( 2x = -4 \).

To finalize, divide both sides by \( 2 \) to solve for \( x \):

• Concluding with \( x = -2 \).

Understanding variable isolation not only helps solve the immediate problem but builds critical thinking skills for exploring more complex mathematical expressions.