Table of Contents
Introduction
To find the solution of an equation, we need to determine the values of the variable that makes the given equation a true statement. Solving an equation is similar to finding out the answer to a puzzle. Any number that makes the equation true is termed the solution of an equation. Hence, it is termed as the answer to a puzzle.
What Is An Equation?
An equation represented by the symbol (=) says that two things are equal. It is represented as
(y – 2) = 4
In other words, it says that what is given on the left side (y – 2) is equal to the right side (4).
Hence, the equation is similar to a statement that says “this equals that”.
What Is Solving An Equation?
A solution to an equation is the value of a variable that makes the statement true when substituted into an equation. Therefore, the process of determining a solution to an equation is known as solving the equation.
How To Determine Whether Number Is A Solution To An Equation?
Following are the steps to determine whether the number is a solution to an equation:
- The first step is to substitute the variable’s value in the given equation.
- The second step is to solve the expression on both sides of the equation.
- The third step is to find out whether the resulting equation is true or not.
If the resulting equation is true, the number is a solution.
If the resulting equation is not true, the number is not a solution.
Example: Find the solution for an equation x + 6 = 11
Solution:
Substituting 5 for x in x + 6 = 11 gives
5 + 6 = 11
11 = 11
LHS = RHS
The number “5” equals both sides of an equation.
Hence, 5 is the solution of a given equation.
Linear Equations
Introduction
A linear equation is an equation whose each term is linear (i.e, the product of a variable and the first power of a variable) or constant (simply the real number). In other words, a linear equation is an algebraic equation through which a straight line is formed when an equation is graphed. Each term in the linear equation is either a constant, a product of a constant, or a single variable. This topic can be understood in a fun way from Cuemath.
What is Known as Linear Equation?
A linear equation, also known as a one-degree equation whose greatest power of the variable is always 1. For example:
- y = 7x – 6
- y + 2x -1 = 0
Linear Equation in One Variable
A linear equation in one variable is expressed in the form of Ax + B = 0, where “x” is the variable and a and b are the constant in the equation. In a linear equation in one variable, the constants a and b should be non-zero real numbers.
Examples:
- 4x + 3 = 15
- 3y + 43 = 8
- 2z + 15 = 82
How To Solve Linear Equations In One Variable?
Following are the steps to solve linear equation in one variable:
- Observe the given linear equation carefully.
- Determine the quantity you need to find out.
- Split the equation into two parts i.e. LHS and RHS.
- Now find out the terms containing constants and variables.
- Transfer all the constants on the RHS of the equation and variables on the LHS of the equation.
- To find the value of the variable, perform algebraic operations on both sides of the equation.
Example: Solve x + 12 = 18
Solutions:
Let us first transfer constants and variables on the LHS and RHS respectively:
x + 12 = 18
x = 18 – 12
x = 6
The value of x is 6.
Linear Equation In Two Variables
A linear equation in two variables is expressed in the form of Ax + By + C = 0. Here a, b, and c are real numbers and a and b are non-zero. For example:
- x + y = 150
- 2x + 3y = 4.5
How To Solve Linear Equations In Two Variables?
There are three methods to solve linear equations in two variables. These are:
- Graphing Method
- Substitution Method
- Elimination Method
Example: Find the solution of the following linear equation in two variables:
2a + 3b = 12
Considering a = 3 and b = 3, we get
2a + 3b = 2(3) + 3(2) = 12
12 = 12
This solution can be written in an ordered pair like (3,2). First, write the value of a and then write the value of b.
Hence, (3,2) is the solution of a given linear equation in two variables.
