What is a linear equation?
Any equation of the first degree is known as a linear equation.
Linear equation in one variable - It is an equation of the form ax + b = 0, a≠ 0, where a and b are constants and x is a variable.
ax + b = 0
is known as the solution or root of the equation ax + b = 0.
Consider the following equations:
1)2x + 1 = 0,
The root of the equation is x = -
3x - 2 = 0,
The equation is x = or it is called the root of the equation.
4x +7= 3
The solution of this equation is 4x = 3 – 7
x =
is the root of the equation.
POINTS TO NOTE
The solution of a linear equation is not affected when
1)the same number is either added or subtracted from both sides of the equation.
2) the same non-zero number is multiplied or divided on both sides of the equation.
Linear Equation in Two Variables
It is an equation of the form ax + by + c = 0, where a and b are real numbers, a ¹ 0, b ¹ 0 and c are constants (real numbers) and x, y are variables. Let us look at the following linear equation in 2 variables
1) x + 2y = 3,
Here x + 2y = 3 can be written as
x + 2y - 3 = 0. Here it is in the form ax + by + c = 0 where a =1, b =2 and c = -3
2) 2x - 3y - 5 = 0
It is in the form ax + by + c = 0 where a =2, b =-3 and c = -5
3) 5x+7y = 4,
Here 5x + 7y = 4 can be written as
5 x + 7y - 4 = 0
It is in the form ax + by + c = 0 where a =5, b =7and c = -4
Solution
A pair of values of x and y satisfying the equation ax + by + c = 0 is known as the solution of the equation.
Example : x = 1, y = 2 is a solution of x + 3y = 7.
Solution of a Linear Equation
We observed that every linear equation in one variable has a unique solution. The solution of a linear equation in two variables is a pair of values (x, y) which satisfy the given equation. Consider an equation 2x + 5y = 17. Here, x = 1 and y = 3 is a solution as when we substitute x = 1 and y = 3 in the equation, we get2x + 5y = 2´1 + 5´ 3 = 2 + 15 = 17
Thus the solution is (1, 3). Similarly (6, 1) is also a solution for the above equation. Try the pair (0, 3). L.H.S = 2´0 + 5´3 = 0 + 15 = 15, which is not 17. Hence (0, 3) is not a solution to the above equation. But this doesn’t mean that there are no more solutions. We can get many solutions in the following way:
Example
Let x = 3 and 2 ´ 3 + 5y = 17 i.e., 5y = 17 – 6 Þ 5y = 11 Þ y = .
.
So there is no end to different solutions of a linear equation in two variables. That is, a linear equation in two variables has infinitely many solutions.
Example :
1) Find four solutions of the equation x + 3y = 7
Let us choose x = 1 then 3y =6 and hence y =2
Therefore x = 1 and y =2 is a solution of the equation x + 3y = 7
If we take x =4 then 3y =3 or y =1
Therefore x = 4 and y =1 is a solution of the equation x + 3y = 7
If we assume x =0 then y = ,is a solution of the equation x + 3y = 7
If we assume y = 0 then x = 7, is a solution of the equation x + 3y = 7
(1, 2), (4, 1), (0, ), (0 ,7) are the four solutions of the infinitely many solutions.
2) Find two solutions of the equation 5x + 3y = 8
Let us assume x=1 then 3y =8 - 5 =3y=3Þ .
x=1 and y = 1 is a solution of the equation 5x + 3y = 8
Let us assume x = 0 then y =
x = 0 and y = is a solution of the equation 5x + 3y = 8.
Hence (1, 1) and (0, ) are the two solutions of the infinitely many solutions.
Graph of Linear Equation in Two Variables
When we draw the graph of the equation ax + by + c = 0, we get a straight line.
Note: The graph of a first degree equation in one or two variables is a (straight) line. Because of this reason we call it a linear equation.
Draw the graph of y = 2x - 1. Use it to find some solutions of the equation. Check from the graph whether x = -1, y = -3 is the solution.
y = 2x – 1
If x = 0; y = 2 × 0 - 1 = 0 - 1= -1
If x = 1; y = 2 × 1 - 1 = 2 - 1 = 1
If x = 2; y = 2 × 2 – 1 = 4 - 1 = 3
We thus have the following points:
x | 0 | 1 | 2 |
y | -1 | 1 | 3 |
Plotting A (0, -1), B (1, 1), C (2, 3), we draw the graph.
We see that the point P (-1, -3) lies on the line, which shows that x = -1 and y = -3 is a solution of the equation y = 2x - 1. Q(, -4) and R (-2, -5) lie on the graph indicating that
x =, y = -4 and x = -2, y = -5 are some other solutions of the equation.
Equations of Lines Parallel to the x-axis and y-axis
We would observe that all the points with y-co-ordinate 0 lie on the x-axis and that with x-co-ordinate 0 lies on the y-axis.
Now we can guess the equation of the line parallel to the x-axis as y = k and y-axis as x = k. Let us draw the graph of the equation x – 2 = 0. According to a linear equation in one variable this equation has one solution x = 2.
Considering it as an equation in two variables, it can be expressed as x + y.0 = 2. This equation has infinitely many solutions as y can take any real number.
Let us draw the graph of the equation y – 3 = 0. According to a linear equation in one variable this equation has one solution y =3.
Considering it as an equation in two variables, it can be expressed as 0.x + y = 3. This equation has infinitely many solutions as x can take any real number.
Summary
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