Problem 9 - Elementary Algebra

Question

Solve the equations. $$ 2 x=32 $$

Step-by-Step Solution

Answer

Question: Solve the equation 2x = 32 and find the value of x.

Step 1: Write down the given equation

First, write down the given equation: $$ 2x = 32 $$

Step 2: Divide by 2

To isolate the variable x, we need to divide both sides of the equation by 2: $$ \frac{2x}{2} = \frac{32}{2} $$

Step 3: Simplify

Now we simplify both sides of the equation: $$ x = 16 $$ The solution to the given equation is: $$ x = 16 $$

Key Concepts

Solving Equations

When solving equations, our main goal is to find the value of the variable that makes the equation true. An equation is like a scale, and we have to keep the balance on both sides.
In this problem, we're given the equation \(2x = 32\). To solve it, we need to find what number \(x\) is.
To do this, we'll perform the same operation on both sides of the equation. This ensures the balance is maintained:

• Identify the operations involved in the equation.
• Perform inverse operations to isolate the variable.

Let's move on to the next section where we discuss the method of isolating variables.

Isolation of Variables

Isolation of the variable is an essential step in solving equations. In our example, the equation \(2x = 32\) shows that \(x\) is multiplied by 2.
The key is to "undo" this operation, so we need to eliminate the 2 next to \(x\).
We can do this by performing the inverse operation. Since multiplication by 2 is used, the inverse would be division by 2:

• Divide both sides of the equation by the number that multiplies the variable, in our case, 2.
• This simplifies the left side of the equation to just \(x\).
• Don't forget to apply the division to the right side too, retaining the equation's balance.

After completing this step, the variable \(x\) will stand alone, leading us to the solution \(x = 16\).

Algebra Basics

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and values in equations and formulas.
At its core, algebra is about finding unknown values by balancing equations.
It's helpful to remember a few basic principles when working with algebra:

• Operations such as addition, subtraction, multiplication, and division can be applied to both sides of an equation as long as they are done equally.
• Always perform operations in reverse order compared to the order of operations (PEMDAS/BODMAS).
• Check your solution by substituting it back into the original equation to see if it holds true.

In our example, solving \(2x = 32\) is relatively straightforward because we deal with a simple linear equation. Mastering these nurturing principles will build a solid foundation for solving more complex problems in the future.