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# Linear Equations

A linear equation contains one or more variables of degree one and a constant term. Some examples of Linear Equations are,

• x + 4 = 12 (A linear equation with a single variable x)
• 2x - 3y = 18  (A linear equation in two variables x and y)
• a1 + a1x1 + a2x2 + ...........+ anxn = 0 (A general linear equation in n variables x1, x2, ....., xn)
A solution of a linear equation in one variable is a numerical value, which satisfies the equation. Or, in other words, gives a true statement, when the variable is replaced by that value in the equation.

A linear equation in one variable will have only one solution, where as linear equations in more than one variable will have infinite number of solutions.
A linear equation in two variables is graphically represented by a straight line, on which account the equations get their name. The stuff here deals briefly on linear equations in one and two variables.

The solution of a single variable linear equation is given by an equivalent equation with the variable isolated on one side of the equation. This is achieved by applying the inverse operations with the support of properties of equality.

 Properties of Equality Addition Property of Equality Adding the same quantity on both sides of an equation results in an equivalent equation.For any real numbers a, b and c, if  a = b, then a + c = b + c. Subtraction Property of Equality Subtracting the same quantity from both the sides of an equation results in an equivalent equation.If a = b, then a - c = b - c. Multiplication Property of Equality Multiplying both the sides of an equation by the same quantity results in an equivalent equation.If a = b, then ac = bc. Division Property of Equality Dividing both sides of an equation by the same quantity results in an equivalent equation.If a = b, then $\frac{a}{c}$ = $\frac{b}{c}$, where c $\neq$ 0.

Examples on Linear Equations:

Given below are some of the examples on linear equations.

Example 1:

Solve 2x - 5 = 9

Solution:

5 is added on both the sides to remove the effect of subtraction by 5.
2x - 5 + 5 = 9 + 5
2x = 14

$\frac{2x}{2}$ = $\frac{14}{2}$

Either side of the equation is divided by 2 to remove the effect of multiplication earlier done.
x = 7
This is the equivalent equation and the solution.

A Linear equation in two variables is represented by a straight line on coordinate plane. A solution of a two variable linear equation is an ordered pair (x, y) satisfying the equation.
Example 2:

Check whether (3,2), (0,3) and (4,0) are solutions of the equation, 3x + 4y = 12.

Solution:

To check (3, 2) is a solution of the equation, we substitute x = 3 and y = 2 in the equation.
3 (3) + 4 (2) = 12
9 + 8 = 12 which is a false statement.
Hence, we conclude (3, 2) is not a solution of the given equation.

Let us check the point (0, 3) in the equation.
3 (0) + 4 (3) = 12
12 = 12   which is a true statement.
Hence, (0, 3) is a solution of the equation 3x + 4y = 12.

Now, let us check the point (4, 0) in the equation.
3 (4) + 4 (0) = 12
12 = 12   which is a true statement.
Hence, (4, 0) is a solution of the equation 3x + 4y = 12.

The graph of the above equation is shown below:

A linear equation in two variables have infinite number of solutions. Any point on the line graph is a solution to the equation. Thus, the straight line graphed displays the complete solution set of the above equation.

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