What are Linear Expressions?
A linear expression is an algebraic expression with the highest exponent of a variable being 1.
A linear expression in one variable is known as a one-variable linear expression. The standard form of a one variable linear expression is Ax + B where x is the variable, A is the coefficient of x, and B is the constant. Here, A can not be zero in any case.
For example: \(3x \text{ }- 1, \text{ }4x, \text{ }97x + 3, 5 \text{ }- 3\) are a few examples of linear expressions.
A linear expression in two variables is known as a two-variable linear expression. The standard form of a two variable linear expression is Ax+By+C where x and y are the variables, A is the coefficient of x, B is the coefficient of y and C is the constant.
For example: \(x + y + 1, x \text{ }- 3y + 14, \text{ }y \text{ }- \text{ }5x – 3\) are some linear expressions in two variables.
Adding Linear Expressions
We can add linear expressions using two different methods.
- Vertical Method
In this method we will first align the like terms vertically and then add them.
For example: Find\(\left( x + 4 \right) + \left( 2x \text{ }- 3 \right)\)
Solution:Align like terms vertically and add.
- Horizontal Method
In this method we use the properties of mathematical operations to group the like terms, simplify them and then add them.
For example:\(\left( x + 4 \right) + \left( 2x \text{ }- 3 \right)\)
Solution: Use the properties of operations to group the like terms and simplify.
\(\left( x + 4 \right) + \left( 2x \text{ }- 3 \right) = x + 4 + 2x \text{ }- 3\) Rewrite the sum
\(= x + 2x + 4 \text{ }- 3 \) Commutative property of addition
\(= \left( x + 2x \right) + \left( 4 \text{ }- 3 \right)\) Group the like terms
\(= 3x + 1\) Add the like terms
The sum is 3x + 1.
Solved Examples
Example 1: Add the linear expressions, (11y – 9) and (5y + 14) by applying the vertical method.
Solution:
Align like terms vertically and add.
The sum is 16y + 5.
Example 2: Add the given linear expressions by applying the horizontal method, \(\left( 4x + 7 \right)\text{ }and\text{ } \left( 7x – 4 \right)\)
Solution:
\(\left( 4x + 7 \right)\text{ }+\text{ } \left( 7x-4 \right)=\text{ }4x + 7 + 7x – 4\) Rewrite the sum
\(= 4x + 7x + 7 – 4\) Commutative Property of Addition
\(= \left( 4x + 7x \right)+\left( 7 – 4 \right)\) Group the like terms
\(= 11x + 3\) Add the like terms
The sum is 11x + 3.
Example 3: Jack was walking around a rectangular park with a width of 2x + 3y-1 meters and a length of x + y + 8 meters. He wants to calculate the total length of the boundary of the rectangular park. Help him in finding the total length.
Solution:
The total length of the boundary of the rectangular park is the perimeter of the park. Calculate the perimeter using the formula for the perimeter of a rectangle.
\(P = 2\left( l+w \right)\) Formula for perimeter of rectangle
Find \(l + w\) then multiply it by 2 to get the perimeter, where \(l = x + y + 8\) meters and \(w = 2x + 3y – 1\) meters.
\(\left( x + y + 8 \right) + \left( 2x + 3y – 1 \right)\) Rewrite the sum
\(= x + y + 8 + 2x + 3y – 1\) Commutative property of addition
\(= \left( x + 2x \right) + \left( y + 3y \right) + \left( 8 – 1 \right)\) Group the like terms
\(= 3x + 4y + 7\) Add the like terms
The sum of length and width is 3x + 4y + 7.
\(P = 2 \left( 3x + 4y + 7 \right)\)
\(= 6x + 8y + 14\)
So, the total length of the boundary of the rectangular park is 6x + 8y + 14 meters.
An algebraic expression is a mathematical expression that consists of variables, coefficients, constants, and mathematical operators.
A linear expression is an algebraic expression in which the highest exponent of a variable is 1.
Nonlinear algebraic expressions are algebraic expressions in which the highest exponent of a variable is greater than 1. Such expressions contain more than one variable with exponents greater than one.
The terms with the same variable of the same exponent are known as like terms.
The terms that do not have either the same variable or the same exponents of the variables are known as unlike terms.
In an algebraic expression:
- A variable is a letter that represents an unknown quantity
- The number associated to a variable is known as the coefficient
- A constant is a number whose value never changes in the expression