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A linear equation in one variable is a basic concept in mathematics. It is used to represent and solve a mathematical equation. A linear equation in one variable is usually expressed in the form of a straight line when represented graphically. It is the simplest way to describe mathematical relationships.
In most cases, the unknown quantity is represented by a variable. These variables are commonly denoted by ‘x’, though other symbols can be used as well. To solve a linear equation in one variable, we gather the variable on one side of the equation while keeping constants on the other side. This step-by-step process makes it easy to find the value of the unknown without making any mistakes.
What is a Linear Equation in One Variable?
A linear equation is an equation where the highest power (or degree) of the variable is exactly one. This means there are no exponents higher than one involved. A linear equation in one variable is a specific type of linear equation that contains only one variable. It has only one solution and it forms a straight line when represented on a graph.
A linear equation in one variable follows the below-given standard form:
ax + b = 0
where ‘a’ and ‘b’ are integers, and ‘x’ is the unknown variable that has only one solution.
An example of linear equation in one variable is discussed below.
Find the x such that if 4 is added to x, it gives 10.
To solve this, we can write the equation as:
x + 4 = 10
Here, x is the unknown number we need to find. This equation can be solved by isolating x on one side:
x = 10 – 4
x = 6
So, the value of x is 6.
In the above-discussed example, the equation x + 4 = 10 is a linear equation in one variable because it has one variable (x) and the highest power of that variable is one.
Also Check: Linear Equations Formula
Solving Linear Equation in One Variable
The general form of a linear equation in one variable is written as:
Ax + B = 0
Here: A is the coefficient of the variable x.
x is the unknown variable.
B is the constant term.
To solve this equation, we need to separate the variables and constants. This is done in two simple steps:
- Move the constant to the other side:
- Subtract or add B from both sides of the equation.
Example: Ax + B = 0
Ax = -B
Now, let’s further solve for x:
Divide both sides of the equation by the coefficient A.
x = -B/A
This step-by-step process gives the solution to the linear equation. By simplifying the equation step by step, you can easily find the value of x.
Also Check: Application of Linear Equations
Balancing a Linear Equation in One Variable
Solving a linear equation is like maintaining balance on a weighing scale, where both sides of the equation must remain equal. You must always remember that whatever operation you perform on one side of the equation must be performed on the other side as well. This is important to maintain that balance between both sides of the equation.
Steps to Solve a Linear Equation in One Variable
Follow the steps below to solve a linear equation in one variable.
- Move the variable term to one side and the constants to the other side by adding or subtracting both sides equally.
- Simplify the constants on each side of the equation.
- Isolate the variable by multiplying or dividing both sides by the coefficient of the variable.
- Simplify and solve to find the value of the variable.
Linear Equation in One Variable vs. Non-Linear Equations
Linear equations in one variable are the simplest form of algebraic equations. These equations have a single degree, meaning the highest power of the variable is one. When plotted on a graph, they form a straight line. Linear equations are easy to solve and have numerous applications in areas like algebra and geometry.
On the other hand, non-linear equations involve variables raised to powers higher than one or involve more complex relationships between variables. These equations do not form straight lines when graphed; instead, they produce curves like circles, parabolas, ellipses, or hyperbolas. Non-linear equations play a critical role in advanced fields like trigonometry, calculus, and geometry, as they describe more complex relationships.
Also Check: Graph of Linear Equations
Examples of Linear Equations:
x = 5
3x + 7 = 9
4x + 2y = 11
Examples of Non-Linear Equations:
Equation of a circle:- x2 + y2 = 25
Equation of an ellipse:- x2/9 + y2/16 =1
Equation of a hyperbola: – x2/16 – y2/25 =1
Linear and non-linear equations serve different purposes and have distinct applications in various mathematical fields. Linear equations are fundamental for simpler, straightforward problems, while non-linear equations handle more complex, multi-variable scenarios.
Important Notes on Linear Equations in One Variable
The following are the important points related to linear equations in one variable:
- The degree of the variable in a linear equation must be exactly one. This means the variable cannot be raised to a higher power (such as squared or cubed).
- The graph of a linear equation in one variable forms a straight line. This line is either horizontal or vertical depending on the structure of the equation.
- The solution to a linear equation remains unchanged if you add, subtract, multiply, or divide the same number on both sides of the equation. This balance ensures that the equation holds true throughout the solving process.
Solved Examples: Linear Equations in One Variable
Example 1: Five years ago, Emma’s age was half of what it will be in 10 years. What is Emma’s current age?
Solution: Let Emma’s current age be x years.
Five years ago, her age was x – 5.
In 10 years, her age will be x + 10.
According to the given information:
x – 5 = 1/2(x + 10)
Multiply both sides by 2 to eliminate the fraction:
2(x – 5) = x + 10
2x – 10 = x + 10
Now, subtract x from both sides:
2x – x = 10 + 10
x = 20
Therefore, Emma’s current age is 20 years.
Example 2: The sum of a number and 7 is equal to three times the number. Find the number.
Solution: Let the unknown number be x.
According to the given information:
x + 7 = 3x
Now, subtract x from both sides:
7 = 3x – x
7 = 2x
Now, divide both sides by 2:
x = 7 / 2
x = 3.5
Therefore, the number is 3.5.
Example 3: A number decreased by 15 is equal to twice the number. Find the number.
Solution: Let the unknown number be x.
According to the given information:
x – 15 = 2x
Now, subtract x from both sides:
-15 = 2x – x
-15 = x
Therefore, the number is -15.
Example 4: The difference between a number and its two-thirds is 12. What is the number?
Solution: Let the unknown number be x.
According to the given information:
x – 2/3x = 12
Multiply the whole equation by 3 to eliminate the fraction:
3(x – 2/3x) = 3 × 12
3x – 2x = 36
x = 36
Therefore, the number is 36.
Linear Equations in One Variable: Practice Questions
- Seven years ago, Sarah’s age was half of what her age will be in 5 years. What is Sarah’s current age?
- The sum of a number and 12 is equal to twice the number minus 8. Find the number.
- Three times a number decreased by 4 equals two times the number increased by 6. Find the number.
- A number increased by 20 is equal to five times the number minus 10. Find the number.
- The difference between four times a number and 7 is equal to three times the number plus 5. Find the number.
FAQs on Linear Equations in One Variable
What is a linear equation in one variable?
A linear equation in one variable is an algebraic equation where the highest power of the variable is one. It is written in the form Ax + B = 0, where A and B are constants, and x is the unknown variable.
How do you solve a linear equation in one variable?
To solve a linear equation in one variable, bring all the variables on one side of the equation by performing operations such as adding, subtracting, multiplying, or dividing both sides by the same number.
What is the graphical representation of a linear equation in one variable?
A linear equation in one variable is represented as a straight line on a graph. Depending on the equation, the line may be horizontal or vertical.
