Example Of A One Step Equation

Mathematics is a fundamental language that unlocks the secrets of the universe. At its core lies algebra, a powerful tool for representing relationships and solving for unknowns. One-step equations, the building blocks of algebraic problem-solving, are the first steps in mastering this essential skill.

Introduction to One-Step Equations

A one-step equation is an algebraic equation that can be solved in only one step. It involves a single variable and a single operation, such as addition, subtraction, multiplication, or division. The goal is to isolate the variable on one side of the equation to determine its value. Understanding these simple equations is crucial because they form the foundation for more complex algebraic problems.

What is an Equation?

Before diving into one-step equations, it's important to understand what an equation is. An equation is a mathematical statement that asserts the equality of two expressions. It always contains an equals sign (=), which indicates that the expressions on either side have the same value. For example:

x + 5 = 10

In this equation, 'x + 5' is the expression on the left side, and '10' is the expression on the right side. The equation is stating that these two expressions are equal.

Basic Concepts in Algebra

To solve one-step equations, a basic understanding of algebraic concepts is required:

• Variable: A variable is a symbol (usually a letter) that represents an unknown value. In the equation above, 'x' is the variable.
• Constant: A constant is a fixed value that does not change. In the equation above, '5' and '10' are constants.
• Coefficient: A coefficient is a number that multiplies a variable. For example, in the expression '3x', '3' is the coefficient.
• Operation: An operation is a mathematical process such as addition, subtraction, multiplication, or division.

Why are One-Step Equations Important?

One-step equations might seem simple, but they are fundamental to understanding algebra and solving more complex problems. Here's why they are important:

• Foundation for Algebra: They lay the groundwork for solving multi-step equations, inequalities, and other algebraic concepts.
• Problem-Solving Skills: They help develop critical thinking and problem-solving skills, which are applicable in various areas of life.
• Practical Applications: They are used in everyday situations, such as calculating expenses, measuring ingredients for a recipe, or determining distances.

Types of One-Step Equations

One-step equations can be categorized based on the operation involved: addition, subtraction, multiplication, or division. Each type requires a specific approach to isolate the variable.

Addition Equations

Addition equations involve adding a constant to a variable. The goal is to isolate the variable by performing the inverse operation, which is subtraction.

Example:

x + 7 = 15

To solve for 'x', subtract '7' from both sides of the equation:

x + 7 - 7 = 15 - 7

x = 8

Explanation:

• The equation states that 'x' plus '7' equals '15'.
• To isolate 'x', we subtract '7' from both sides of the equation.
• This cancels out the '+ 7' on the left side, leaving 'x' by itself.
• The result is 'x = 8', which means the value of 'x' that satisfies the equation is '8'.

Subtraction Equations

Subtraction equations involve subtracting a constant from a variable. To isolate the variable, perform the inverse operation, which is addition.

Example:

y - 5 = 3

To solve for 'y', add '5' to both sides of the equation:

y - 5 + 5 = 3 + 5

y = 8

Explanation:

• The equation states that 'y' minus '5' equals '3'.
• To isolate 'y', we add '5' to both sides of the equation.
• This cancels out the '- 5' on the left side, leaving 'y' by itself.
• The result is 'y = 8', which means the value of 'y' that satisfies the equation is '8'.

Multiplication Equations

Multiplication equations involve multiplying a variable by a constant. To isolate the variable, perform the inverse operation, which is division.

Example:

4z = 20

To solve for 'z', divide both sides of the equation by '4':

4z / 4 = 20 / 4

z = 5

Explanation:

• The equation states that '4' times 'z' equals '20'.
• To isolate 'z', we divide both sides of the equation by '4'.
• This cancels out the '4' on the left side, leaving 'z' by itself.
• The result is 'z = 5', which means the value of 'z' that satisfies the equation is '5'.

Division Equations

Division equations involve dividing a variable by a constant. To isolate the variable, perform the inverse operation, which is multiplication.

Example:

a / 3 = 6

To solve for 'a', multiply both sides of the equation by '3':

(a / 3) * 3 = 6 * 3

a = 18

Explanation:

• The equation states that 'a' divided by '3' equals '6'.
• To isolate 'a', we multiply both sides of the equation by '3'.
• This cancels out the '/ 3' on the left side, leaving 'a' by itself.
• The result is 'a = 18', which means the value of 'a' that satisfies the equation is '18'.

Step-by-Step Guide to Solving One-Step Equations

Solving one-step equations involves a systematic approach. Here's a step-by-step guide to help you solve these equations:

1. Identify the Variable: Determine the variable you need to solve for.
2. Identify the Operation: Determine the operation being performed on the variable (addition, subtraction, multiplication, or division).
3. Perform the Inverse Operation: Perform the inverse operation on both sides of the equation to isolate the variable.
4. Simplify: Simplify both sides of the equation to find the value of the variable.
5. Check Your Solution: Substitute the value of the variable back into the original equation to verify that it is correct.

Example 1: Solving an Addition Equation

Equation:

x + 9 = 16

Steps:

1. Identify the Variable: The variable is 'x'.

2. Identify the Operation: The operation is addition (+ 9).

3. Perform the Inverse Operation: Subtract '9' from both sides of the equation:

   x + 9 - 9 = 16 - 9

4. Simplify:

   x = 7

5. Check Your Solution: Substitute '7' for 'x' in the original equation:

   7 + 9 = 16

   16 = 16 (The solution is correct)

Example 2: Solving a Subtraction Equation

Equation:

y - 4 = 7

Steps:

1. Identify the Variable: The variable is 'y'.

2. Identify the Operation: The operation is subtraction (- 4).

3. Perform the Inverse Operation: Add '4' to both sides of the equation:

   y - 4 + 4 = 7 + 4

4. Simplify:

   y = 11

5. Check Your Solution: Substitute '11' for 'y' in the original equation:

   11 - 4 = 7

   7 = 7 (The solution is correct)

Example 3: Solving a Multiplication Equation

Equation:

5z = 30

Steps:

1. Identify the Variable: The variable is 'z'.

2. Identify the Operation: The operation is multiplication (5 * z).

3. Perform the Inverse Operation: Divide both sides of the equation by '5':

   5z / 5 = 30 / 5

4. Simplify:

   z = 6

5. Check Your Solution: Substitute '6' for 'z' in the original equation:

   5 * 6 = 30

   30 = 30 (The solution is correct)

Example 4: Solving a Division Equation

Equation:

b / 2 = 9

Steps:

1. Identify the Variable: The variable is 'b'.

2. Identify the Operation: The operation is division (b / 2).

3. Perform the Inverse Operation: Multiply both sides of the equation by '2':

   (b / 2) * 2 = 9 * 2

4. Simplify:

   b = 18

5. Check Your Solution: Substitute '18' for 'b' in the original equation:

   18 / 2 = 9

   9 = 9 (The solution is correct)

Common Mistakes to Avoid

When solving one-step equations, it's important to avoid common mistakes that can lead to incorrect solutions. Here are some mistakes to watch out for:

• Not Performing the Same Operation on Both Sides: Always perform the same operation on both sides of the equation to maintain balance.
• Incorrectly Identifying the Inverse Operation: Make sure you are using the correct inverse operation (addition for subtraction, subtraction for addition, multiplication for division, and division for multiplication).
• Forgetting to Simplify: Always simplify both sides of the equation after performing the inverse operation.
• Not Checking the Solution: Always check your solution by substituting it back into the original equation to verify that it is correct.

Example of an Incorrect Solution

Equation:

x - 3 = 5

Incorrect Steps:

1. Add '3' to only one side of the equation:

   x - 3 + 3 = 5 (Incorrect)

2. Simplify:

   x = 5 (Incorrect)

Correct Steps:

1. Add '3' to both sides of the equation:

   x - 3 + 3 = 5 + 3

2. Simplify:

   x = 8

Explanation:

The mistake was adding '3' only to the left side of the equation. To maintain balance, the same operation must be performed on both sides.

Practical Applications of One-Step Equations

One-step equations are not just abstract mathematical concepts; they have practical applications in everyday life. Here are some examples:

• Calculating Expenses: If you know that your total expenses for the week are $100, and you spent $30 on groceries, you can use a one-step equation to find out how much you spent on other items:

  30 + x = 100

  Solving for 'x' gives you:

  x = 70

  So you spent $70 on other items.

• Measuring Ingredients for a Recipe: If a recipe calls for 2 cups of flour, and you only want to make half the recipe, you can use a one-step equation to find out how much flour you need:

  x = 2 / 2

  Solving for 'x' gives you:

  x = 1

  So you need 1 cup of flour.

• Determining Distances: If you know that you have traveled 50 miles and you are halfway to your destination, you can use a one-step equation to find out the total distance:

  50 = x / 2

  Solving for 'x' gives you:

  x = 100

  So the total distance is 100 miles.

• Calculating Time: If you know that you need to bake a cake for 45 minutes, and you have already baked it for 20 minutes, you can use a one-step equation to find out how much more time you need:

  20 + x = 45

  Solving for 'x' gives you:

  x = 25

  So you need to bake the cake for another 25 minutes.

Real-World Examples

Let's explore some more detailed real-world examples to illustrate how one-step equations are used in various scenarios.

Example 1: Calculating the Cost of a Shirt

Scenario:

You want to buy a shirt that is on sale for 25% off. The sale price is $15. What was the original price of the shirt?

Solution:

Let 'x' be the original price of the shirt. The sale price is 75% of the original price (100% - 25% = 75%). So, the equation is:

0.75x = 15

To solve for 'x', divide both sides by 0.75:

x = 15 / 0.75

x = 20

Therefore, the original price of the shirt was $20.

Example 2: Dividing Pizza Slices

Scenario:

You have a pizza that is cut into 12 slices. You want to divide the pizza equally among 4 friends. How many slices does each friend get?

Solution:

Let 'x' be the number of slices each friend gets. The equation is:

4x = 12

To solve for 'x', divide both sides by 4:

x = 12 / 4

x = 3

Therefore, each friend gets 3 slices of pizza.

Example 3: Determining the Number of Students in a Class

Scenario:

Half of the students in a class are girls. If there are 15 girls in the class, how many students are there in total?

Solution:

Let 'x' be the total number of students in the class. The equation is:

x / 2 = 15

To solve for 'x', multiply both sides by 2:

x = 15 * 2

x = 30

Therefore, there are 30 students in the class.

Tips and Tricks for Mastering One-Step Equations

Mastering one-step equations requires practice and understanding of the underlying concepts. Here are some tips and tricks to help you improve your skills:

• Practice Regularly: The more you practice, the more comfortable you will become with solving one-step equations.
• Understand the Concepts: Make sure you understand the concepts of variables, constants, and operations.
• Use Inverse Operations: Always use the correct inverse operation to isolate the variable.
• Check Your Solutions: Always check your solutions by substituting them back into the original equation.
• Break Down Complex Problems: If you encounter a complex problem, break it down into smaller, more manageable steps.
• Seek Help When Needed: Don't hesitate to ask for help from a teacher, tutor, or friend if you are struggling.
• Use Online Resources: There are many online resources available, such as videos, tutorials, and practice problems, that can help you learn and practice solving one-step equations.

Conclusion

One-step equations are the foundation of algebra and are essential for developing problem-solving skills. By understanding the basic concepts, following the step-by-step guide, avoiding common mistakes, and practicing regularly, you can master one-step equations and build a strong foundation for more advanced algebraic concepts. Remember to always check your solutions and seek help when needed. With dedication and practice, you can become proficient in solving one-step equations and apply these skills to various real-world scenarios.