Linear Equations
What is a linear equation: An equation is a condition on a variable. A variable takes on different values; its value is not fixed. Variables are denoted usually by letter of alphabets, such as x, y , z , l , m , n , p etc. From variables we form expression.
Linear equation in one variable: These are the type of equation which have unique (i.e, only one and one ) solution.
For example: 2 x + 5 = 0 is a linear equation in one variable.
Root of the equation is $\frac{-5}{2}$
Example 1: Convert the following equation in statement form.
x - 5 = 9
Solution :
5 taken from x gives 9
So x = 9 + 5 = 14
Hence, x = 14
For verification of the statement,
x - 5 = 9
14 - 5 = 99 = 9 So left hand side value is equal to right hand side value.
Hence the value of x determined is correct .
You can try out some more examples from linear equations worksheets
Linear equation in two variable: An equation which can be put in the form ax+by+c=0, where a, b, and c are real numbers, and a and b are not zero, is called linear equation in two variables.
For example: 3 x + 4 y = 8 which is a equation in two variables.
Summary:
A linear equation in two variable has infinitely many solutions.The graph of every linear equation in two variable is a straight line.
Every point on the graph of a linear equation in two variable is a solution of the linear equation.
An equation of the type y = mx represents a line passing through the origin.
Below are the methods for solving linear equations in one variable:
Method 1: Isolate the variable: In this method we will isolate the variable on one side and number on other sides.
Steps and example for solving equation:
Example 1: solve 2x + 3 =15
Solution 1: Given equation is: 2x + 3 = 15
Step 2: Subtract 3 from both side
2x + 3 - 3 = 15 - 3
2 x =12
Step3: Isolate the variable by dividing 2 to both side
$\frac{2x}{2}$ = $\frac{12}{2}$
x = 6
Solution is 6
Method 2: Graph method for linear equation in two variable.
The graph of every linear equation in two variables is a straight line. Every point on the graph of a linear equation is a two variables is a solution of the linear equation. moreover, every solution of the linear equation is a point on the graph of the linear equation
Example 1: Solve graphically y + 2x =6
Solution 1: Given equation: y + 2x = 6
x | 0 | 1 | 2 | 3 | -1 | -2 |
y | 6 | 4 | 2 | 0 | 8 | 10 |
The graph is given by:
The equation have infinitely many solutions.
Below are few examples of equations which will help you to understand better how to solve linear equations:
Example of linear equations in one variable:
Example 1: Solve 3n + 7 = 25
Solution1: Given equation is 3n + 7 = 25
Step1: Subtract 7 from both sides
3n + 7 - 7 = 25 - 7
3n = 18
Step 2: Divide both side by 3
$\frac{3n}{3}$ = $\frac{18}{3}$
n = 6
Solution1: n = 6
Example of algebra linear equations in two variables
Example2: Find two different solutions of the equations.
(i) 4x + 3y = 12
(ii) 2x + 5y = 0
Solution 1: Let us take x=0, we get 3y = 12
i.e , y =4, So (0, 4 ) is a solution of the given equation. Similarly, by taking y=0 , we get x = 3, thus , (3, 0) is also a solution.
Two different solutions are (0, 4) and (3, 0)
Solution 2: Let us take x = 0, we get 5y = 0, i.e y = 0.
So (0, 0) is a solution of a given equation.To get another solution, take x = 1, so corresponding value of y is $\frac{-2}{5}$. So (1, $\frac{-2}{5}$) is second solution.
Two different solutions are (0, 0) and (1, $\frac{-2}{5}$).
Here is an example of linear equations word problems:
Example: The sum of three times a number is and 11 is 32. Find the number.
Let's see how to do linear equations.
Solution: let the unknown number is taken to be x,Then three times the number is 3x and the sum of 3 x and 11 is 32. That is 3 x + 11 = 32
We get an equation 3x + 11 = 32
Now, Subtract 11 from both side
3x + 11 - 11 = 32 - 11
3x = 21
Divide both side by 3
$\frac{3x}{3}$ = $\frac{21}{3}$
x = 7
The required number is 7.
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