Linear Equation in One Variable

Linear equations in 1 variable are the foundation of algebra, providing an easy yet powerful way for solving unknown quantities. Solving a linear equation in one variable is like finding a missing piece of a puzzle. Here, the missing piece of the puzzle is the variable that needs to be found. The equations are called linear because, when graphed, they form a straight line in a coordinate plane. Understanding these equations can greatly help in solving various problems which include calculating costs, distances, or unknown values. 

1.0Linear Equation in One Variable Definition

The Single Variable Linear Equations are a type of algebraic equation which involves only one unknown variable (x, y, a, b …) with the highest degree or power of the variable being 1. In other words, the power of the variable in the linear form doesn’t rise greater than one. The linear equations are solved when written in their standard form. Which is: 

ax+b=0

Here, 

• x is the variable (the unknown value we are trying to solve for).
• a is the coefficient of x, a constant number that multiplies the variable.
• b is the constant term, a fixed number that is either added or subtracted in the equation.
• a ≠ 0 because if “a” were zero, the equation would no longer involve x, thus not being a linear equation.

2.0Linear Equation in One Variable Solution

In general, a linear equation in one variable has only one solution. Solutions represent the values of unknown variables; the more the solutions, the more the values of the unknown variable. The only solution represents the uniqueness of these equations. For example: 

3x+4=10

3x=6

x=2

The reason that a linear equation in a single variable usually just has one solution is in the equation itself. So, where higher order equations (for example, quadratic equations), a variable x could hold several values fulfilling the equation, as we have seen with multiple values of x satisfying a quadratic equation, through linear equations, only set one solution is allowed.

3.0Solving Linear Equation in One Variable:

To Solve a linear equation in one variable, follow these few easy steps:

Step 1: The first step is to identify that the equation is in the standard form of ax + b = 0

Step 2: Separation of variable Makes the variable on one side and constants on the other side with the help of arithmetic operations.

Step 3: Rewrite the equation by combining like terms and simplifying both sides to set it up for solving.

Step 4: Solve for the variable: Perform the necessary operations (addition, subtraction, multiplication, or division) to isolate x. 

4.0Linear Equation in One Variable Graph: 

A linear equation in one variable is usually represented on a number line, as it involves only one variable that needs to be solved. But as a linear equation in one variable has only one unknown variable, its graph is not a simple line as seen in the two-variable case. Rather than being a whole graph, the linear equation in one variable is shown on the number line at the value of x, which makes the equation correct like this: 

5.0Linear Equation in One Variable Example

Problem 1: Solve the following equation for equation in one variable:

$\frac{2x-3}{4}+\frac{3x+5}{2}=5$

Solution:

According to the question: $\frac{2x-3}{4}+\frac{3x+5}{2}=5$

Taking LCM, 

$\frac{2x-3+2(3x+5)}{4}=5$

2x-3+6x+10=5

8x=5-7

x=-2/8=-1/4

Problem 2: Sarah is 5 years older than twice her brother's age. The sum of their ages is 35 years. How old is Sarah?

Solution:

According to the questions: 

Let the age of Sarah’s brother = x, 

Hence, the age of Sarah = 2x+5

x+2x+5=35

3x=30

x=10

Age of Sarah = 2(10)+5 = 20+5 = 25 years.

Problem 3: A company produces and sells a product. The fixed cost of producing the product is INR 300, and the variable cost per unit is INR 5. The company sells each product for INR 20. Write a linear equation that represents the company’s total revenue and total cost, and then graph the equations to find the break-even point (where total revenue equals total cost).

Solution:

Let the total number of products = x 

Total revenue earned = 20x

Total cost spent by the company to produce products = 300 + 5x

20x=300+5x

15x=300

x=20

Hence, the total number of products = 20

Worksheet for Linear Equations in One Variable

Q1: Solve for x: 2(x−3)+4(x+2)=16

Q2: A number is decreased by 7 and the result is equal to 3 times the number. What is the number?

Q3: The perimeter of a rectangle is 50 meters. The length of the rectangle is 5 meters, more than twice the width. Find the dimensions of the rectangle.

Q4: A car rental service charges a fixed fee of INR 30 plus INR 0.50 per mile driven. If a customer drove 100 miles, how much did the customer pay?