Understanding One-Variable Equations

A one-variable equation is an equation that contains only one variable (often represented by letters like x, y, etc.). The goal is to find the value of the variable that makes the equation true.

Basic Structure

A one-variable equation typically looks like this:

ax + b = c

Where: x is the variable.

a and b are constants (numbers).

c is also a constant.

Example 1: Simple One-Step Equation

Let’s start with a simple equation: x + 5 = 12

Steps to Solve:

1. Isolate the variable: To find x, we need to get x alone on one side of the equation. In this case, we subtract 5 from both sides of the equation.
2. x + 5 – 5 = 12 – 5
3. Simplify: This simplifies to: x = 7

So, the solution is x = 7.

Verification: Plug x = 7 back into the original equation to ensure it’s correct: 7 + 5 = 12 Since both sides are equal, the solution is correct.

Example 2: Two-Step Equation

Consider the equation: 2x – 3 = 7

Steps to Solve:

1. Isolate the term with the variable: First, add 3 to both sides to move the constant term.
2. 2x – 3 + 3 = 7 + 3
3. Simplify: 2x = 10
4. Solve for the variable: Now, divide both sides by 2 to isolate x.
5. 2x/2 = 10/2
6. x = 5

Verification: Plug x = 5 back into the original equation: 2(5) − 3 = 10 – 3 = 7

The solution x = 5 is correct.

Example 3: Equation with Division

Let’s solve the equation: 4x + 2 = 5

Steps to Solve:

1. Isolate the term with the variable: Subtract 2 from both sides. 4x + 2 – 2 = 5 – 2
2. Simplify: 4x = 3
3. Solve for the variable: Divide both sides by 4 to get 4x/4 = 3/4
4. x = 3/4

Verification: Plug x = 3/4 back into the original equation: 4(3/4) + 2 = 3 + 2 = 5

The solution x = 3/4 is correct.

Example 4: Equation with a Negative Coefficient

Let’s solve: -3x + 4 = 1

Steps to Solve:

1. Isolate the term with the variable: Subtract 4 from both sides. -3x + 4 – 4 = 1 – 4
2. Simplify: -3x = -3
3. Solve for the variable: Divide both sides by -3. x =−3/−3​
4. x = 1

Verification: Plug x = 1 back into the original equation: -3(1) + 4 = -3 + 4 = 1

The solution x = 1 is correct.

Let’s go through a few more examples of one-variable equations, each with different levels of complexity.

Example 5: Multiplication and Addition

Solve the equation: 3x + 7 = 19

Steps to Solve:

1. Isolate the term with the variable: Subtract 7 from both sides. 3x + 7 – 7 = 19 – 7
2. Simplify: 3x = 12
3. Solve for the variable: Divide both sides by 3. 3x/3 = 12/3
4. ​ x = 4

Verification: Substitute x = 4 back into the original equation: 3(4) + 7 = 12 + 7 = 19

The solution x = 4 is correct.

Example 6: Division and Subtraction

Solve the equation: 5x – 2 = 3

Steps to Solve:

1. Isolate the term with the variable: Add 2 to both sides. 5x – 2 + 2 = 3 + 2
2. Simplify: 5x = 5
3. Solve for the variable: Divide both sides by 5.
4. 5x/5 = 5/5
5. x = 1

Verification: Substitute x = 1 back into the original equation: 5x – 2 = 5 – 2 = 3

The solution x = 1 is correct.

Example 7: Negative Variable

Solve the equation: -4x + 6 = -10

Steps to Solve:

1. Isolate the term with the variable: Subtract 6 from both sides. 4x + 6 – 6 = -10 – 6
2. Simplify: – 4x = -16
3. Solve for the variable: Divide both sides by 4.
4. 4x/4=16/4​
5. x = 4
6. Verification: Substitute x = 4 back into the original equation: -4(4) + 6 = -16 + 6 = -10
7. The solution x = 4 is correct.

Example 8: Equation with Fractions

Solve the equation: (2x/3) + 4 = 6

Steps to Solve:

1. Isolate the term with the variable: Subtract 4 from both sides. (2x)/3 + 4 – 4 = 6 – 4
2. Simplify: 2x/3 = 2
3. Solve for the variable: multiply 3 by 2 to eliminate the fraction. 2x = 3 x 2
4. 2x = 6
5. x = 6/2 =3

Verification: Substitute: x = 3 back into the original equation: 2(3)/3 + 4 = 2 + 4 = 6

The solution x = 3 is correct.

Example 9: Solving for a Negative Variable

Solve the equation: -3x + 5 = 2

Steps to Solve:

1. Isolate the term with the variable: Subtract 5 from both sides. -3x + 5 – 5 = 2 – 5
2. Simplify: -3x = -3
3. Solve for the variable: Divide both sides by 3 to eliminate the negative sign. x = 1

Verification: Substitute x = 1 back into the original equation: -3(1) + 5 = -3 + 5 = 2

The solution x = 1 is correct.

Example 10: Equation with Variable on Both Sides

Solve the equation: 2x + 3 = x + 9

Steps to Solve:

1. Move the variable terms to one side: Subtract x from both sides to get all the x terms on one side.
2. 2x – x + 3 = x – x + 9
3. Simplify: x + 3 = 9
4. Solve for the variable: Subtract 3 from both sides.
5. x + 3 – 3 = 9 – 3
6. x = 6

Verification: Substitute x = 6 back into the original equation: 2(6) + 3 = 12 + 3 = 15

6 + 9 = 15

Since both sides equal 15, the solution x = 6 is correct.

These examples show different scenarios where one-variable equations can be solved by performing operations like addition, subtraction, multiplication, or division to isolate the variable and find its value.

Conclusion

One-variable equations involve finding the value of a variable that makes the equation true. The process usually involves isolating the variable by performing operations like addition, subtraction, multiplication, or division on both sides of the equation. Each step is about maintaining the balance of the equation while simplifying it to solve for the unknown variable.