Solving One-Step Equations

Solving one-step equations is a fundamental skill in algebra that lays the groundwork for more complex problem-solving. In this article, we will explore the various methods used to solve these simple equations—specifically how to use addition, subtraction, multiplication, and division to isolate the variable. By the end of this article, you’ll not only be able to solve one-step equations confidently but also appreciate the elegance of using balanced operations.

What is a One-Step Equation?

A one-step equation is an equation that can be solved with a single operation. The general form of a one-step equation can be expressed as \( x + a = b \), \( x - a = b \), \( ax = b \), or \( \frac{x}{a} = b \). Here, \( x \) represents the variable we are trying to solve for, while \( a \) and \( b \) are constants.

To solve these equations, our goal is to isolate the variable (\( x \)) on one side of the equation, making it possible to find its value.

Solving One-Step Equations by Addition

Let’s start with addition. One-step equations that involve addition typically look like this:

\[ x + a = b \]

To isolate \( x \), we need to subtract \( a \) from both sides of the equation. This operation is based on the principle of equality: what we do to one side, we must do to the other.

Example 1: Solve \( x + 5 = 12 \)

  1. Start with the equation: \[ x + 5 = 12 \]

  2. Subtract 5 from both sides: \[ x + 5 - 5 = 12 - 5 \]

  3. Simplifying both sides gives us: \[ x = 7 \]

Thus, the solution to the equation \( x + 5 = 12 \) is \( x = 7 \).

Example 2: Solve \( x + 10 = 25 \)

  1. Start with the equation: \[ x + 10 = 25 \]

  2. Subtract 10 from both sides: \[ x + 10 - 10 = 25 - 10 \]

  3. This simplifies to: \[ x = 15 \]

So, the solution is \( x = 15 \).

Solving One-Step Equations by Subtraction

Next up is solving equations that involve subtraction. An equation of this form generally looks like:

\[ x - a = b \]

To isolate \( x \), you need to add \( a \) to both sides.

Example 3: Solve \( x - 3 = 4 \)

  1. Start with the equation: \[ x - 3 = 4 \]

  2. Add 3 to both sides: \[ x - 3 + 3 = 4 + 3 \]

  3. Simplifying gives us: \[ x = 7 \]

Thus, the solution to \( x - 3 = 4 \) is \( x = 7 \).

Example 4: Solve \( x - 8 = 0 \)

  1. Start with the equation: \[ x - 8 = 0 \]

  2. Add 8 to both sides: \[ x - 8 + 8 = 0 + 8 \]

  3. This simplifies to: \[ x = 8 \]

So, the solution is \( x = 8 \).

Solving One-Step Equations by Multiplication

In cases where multiplication is involved, you will encounter equations of the form:

\[ ax = b \]

Here, to isolate \( x \), you need to divide both sides by \( a \).

Example 5: Solve \( 3x = 12 \)

  1. Start with the equation: \[ 3x = 12 \]

  2. Divide both sides by 3: \[ \frac{3x}{3} = \frac{12}{3} \]

  3. This simplifies to: \[ x = 4 \]

Therefore, the solution to \( 3x = 12 \) is \( x = 4 \).

Example 6: Solve \( -5x = 15 \)

  1. Start with the equation: \[ -5x = 15 \]

  2. Divide both sides by -5: \[ \frac{-5x}{-5} = \frac{15}{-5} \]

  3. This simplifies to: \[ x = -3 \]

Thus, the solution is \( x = -3 \).

Solving One-Step Equations by Division

Lastly, let’s consider equations that involve division, typically structured as:

\[ \frac{x}{a} = b \]

To isolate \( x \), you will need to multiply both sides by \( a \).

Example 7: Solve \( \frac{x}{4} = 3 \)

  1. Start with the equation: \[ \frac{x}{4} = 3 \]

  2. Multiply both sides by 4: \[ 4 \times \frac{x}{4} = 4 \times 3 \]

  3. This simplifies to: \[ x = 12 \]

Therefore, the solution to \( \frac{x}{4} = 3 \) is \( x = 12 \).

Example 8: Solve \( \frac{x}{-2} = 5 \)

  1. Start with the equation: \[ \frac{x}{-2} = 5 \]

  2. Multiply both sides by -2: \[ -2 \times \frac{x}{-2} = -2 \times 5 \]

  3. This simplifies to: \[ x = -10 \]

So, the solution is \( x = -10 \).

Summary

As we have seen, solving one-step equations can be achieved through systematic operations: addition, subtraction, multiplication, or division. Remember that the key to balancing an equation lies in performing the same operation on both sides.

These fundamental skills not only prepare you for more advanced algebraic concepts but also enhance your logical thinking abilities. Whether you are isolating \( x \) in an equation or checking your solutions, these techniques help build a strong mathematical foundation. Practice makes perfect, so try solving various one-step equations until you feel comfortable with each method!

Now that you've learned how to tackle one-step equations confidently, you can apply these principles to more complex algebraic challenges as you continue your mathematical journey. Happy solving!