Problem 34 - Prealgebra

Question

Write an example of an equation with variables on each side. State the steps you would use to isolate the variable.

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Write the Equation

Let's consider the equation \(3x + 5 = 2x + 10\). This equation has variables on both sides of the equation.

Step 2: Move Variable Terms to One Side

To isolate the variable, you need to move all terms with the variable to one side. Subtract \(2x\) from both sides to keep the equation balanced: \(3x + 5 - 2x = 2x + 10 - 2x\), which simplifies to \(x + 5 = 10\).

Step 3: Move Constant Terms to Opposite Side

Now, move constant terms to the other side of the equation by subtracting 5 from both sides: \(x + 5 - 5 = 10 - 5\), resulting in \(x = 5\).

Step 4: Verify the Solution

Substitute \(x = 5\) back into the original equation to verify: \(3(5) + 5 = 2(5) + 10\) becomes \(15 + 5 = 10 + 10\), which simplifies to \(20 = 20\). This shows the solution is correct.

Key Concepts

Understanding Variables

Variables are symbols used in equations to represent unknown numbers or values. In mathematical expressions, these variables are often represented by letters like \(x\), \(y\), or \(z\). They allow us to generalize and solve problems where specific values are not initially known. For example, in the equation \(3x + 5 = 2x + 10\), \(x\) is a variable.

By using variables, one can set up all kinds of equations to solve different problems, varying from simple algebraic ones to more complex scenarios like in calculus or physics. Variables are essential because they provide flexibility and allow us to model real-world situations with mathematical expressions.

Isolating Variables in Equations

Isolating variables means rearranging an equation so that one variable is on one side of the equation, often alone. This process allows us to find the value of that variable. For instance, consider the equation \(3x + 5 = 2x + 10\).

Here are some steps to isolate \(x\):

• Identify the variable terms: In \(3x + 5 = 2x + 10\), the variable terms are \(3x\) and \(2x\).
• Move all variable terms to one side: Subtract \(2x\) from both sides to get \(x + 5 = 10\). This balances the variables more efficiently and keeps the equation clear.
• Remove constants: Subtract \(5\) from each side to get \(x = 5\). Now \(x\) is isolated, providing an answer to the equation.

Isolating the variable helps simplify equations and makes it easier to solve for the unknown value. This step is crucial in solving algebraic equations and understanding deeper mathematical principles.

Balancing Equations

Balancing equations is an essential skill in algebra and involves maintaining equality on both sides of the equation as changes are made. When solving an equation, each operation performed on one side must also be performed on the other side to keep the equation balanced.

In our example, \(3x + 5 = 2x + 10\), consider how balancing occurs:

• When \(2x\) is subtracted from both sides, it affects both sides in the same way, so the equation remains balanced, giving \(x + 5 = 10\).
• Similarly, when \(5\) is subtracted from both sides, \(x\) alone equals \(5\), maintaining the balance. Thus, \(x = 5\) is a balanced state.

Balancing equations ensures that whatever you do to one side, you do to the other, preserving the equality. This careful maintenance of equality is what allows us to manipulate and solve equations logically and correctly.