Students can go through AP 9th Class Maths Notes Chapter 4 Linear Equations in Two Variables to understand and remember the concepts easily.
Class 9 Maths Chapter 4 Notes Linear Equations in Two Variables
→ Linear equation : An equation whose degree of variable is one is called linear equation.
Ex : ax + b = 0 here ‘x’ is only one variable and its degree is 1. a, b are real numbers. So, it is called lihear equation of one variable. It will have only one solution which can be represented by unique point on number line.
→ An equation with two variables and degrees of both variables is equal to 1, then such an equation is called linear equation with two variables.
For example, ax + by + c = 0 where x, y are two variables with degree 1 and a, b, c are real numbers.
→ Solution: (Procedure of getting solutions) related to these linear equation with two variables are learnt in this chapter.
→ It is important to note, the solution of linear equation (any degree) won’t be affected when
- the same number is added or subtractced from both sides of an equation.
- or multiplied / divided both sides of equation by the same non-zero real number.
Ex:
(i) ax + b = 0, then ax + b ± c = 0 ± c (‘c’ is added / subtracted)
(ii) ax + by + c = 0, then ax + by + c ± P = 0 ± P.
(iii) ax + b = 0, then P(ax + b) = 0, (P) = 0
(or) \(\frac{a x+b}{P}\) = \(\frac{1}{\mathrm{P}}\) (ax + b) = 0 (\(\frac{1}{\mathrm{P}}\)) = 0
iv) ax + by + c = d, then P (ax + by + c)
= d (P) or \(\frac{1}{\mathrm{P}}\) (ax + by + c) = d (\(\frac{1}{\mathrm{P}}\))
→ In general we use the letters x, y, p, q, r, s, t, u,… for variables in equations.
→ And a, b, c, d, e,… for constants / coefficients. The general form of a linear equation in two variables (x, y) is ax + by + c = 0.
(Where a, b are non-zero real numbers, c is any real number.)
This can be written in any form of following,
ax + by = -c (or) ax = -(by + c) (or) ax + c = -by etc.
→ Solution of a linear equation of one variable is only one. Hence it can be represented by a unique point on a number line.
Ex : 2x – 4 = 6 hence x = 5
2p + 8 = 0 hence p = -4
x – π = 1 hence x = 1 + π
all are only one solution.
But in the case of linear equation with two variables, solutions are many.
For example,
x + y = 1000
x = 1, y = 999 is a solution,
x = 2, y = 998 is a solution,
x = 3, y = 997 is a solution,
x = 0, y = 1000 is a solution,
x = 1.1, y = 998.9 is a solution,
x = -1000, y = 2000 is a solution.
- Thus we can write unlimited solutions.
- So every solution is a point on a line(need not be number line).
- Then ail above solutions represent different points.
- Then by joining all above points we get a straight line.
- So, this straight line represents our linear equation.
- And all points on that line are solutions to our linear equation in two variables.
- The graph of a linear equation in two variables is a straight line.
- It is clear to us that X-coordinate of all points on Y-axis is zero.
- Hence X = 0 is the equation which represents Y-axis, it means equation of Y- axis is X = 0.
- Similarly, y-coordinate of all points on X-axis is zero.
- Hence Y = 0 is the equation which represents X-axis. It means, equation of X- axis is Y = 0.
- In the same manner, we can understand X = k is the equation which represents a line parallel to Y-axis at a distance of ’k’ units from origin.
- Since X-coordinate of ail points on a line parallel to Y-axis is same. And its value is equal to distance from origin point.
- Hence Y = k is the equation, which represents a line parallel to X-axis and at a distance of ‘k’ units from origin.
Since Y-coordinate of all points on a line parallel to X-axis are same. And equal to the distance from origin point. - An equation of the type y = mx represents a line passing through the origin.
- y = mx + c (c ≠ 0) is an equation which represents a line that doesn’t pass through the point origin.
Checking whether the given is a solu¬tion of particular linear equation in two . variables: - To check, we have to. substitute the given values in the place of variables in the equation, after substitution, if the equality holds good, then the given would be a solution. Otherwise it won’t
be a solution.
Ex: Check (1, 4) (9, 6),‘ (3, 3) are solutions of the linear equation x + 2y = 9.
Solution:
Checking for. (1, 4), put x = 1, y = 4 in given equation,
x + 2y = 9 ⇒ 1 + 2 (4) = 9
⇒ 1 + 8 = 9 ⇒ 9 = 9
LHS = RHS
Hence (1, 4) is a solution of x + 2y = 9.
Now checking for (9, 6)
Put x = 9, y = 6 in given equation
x + 2y = 9 ⇒ 9 + 2 (6) = 9
f ⇒ 9 + 12 = 9 ⇒ 21 ≠ 9
LHS ≠ RHS
Hence (9, 6) is not a solution to given x + 2y = 9.
And now checking for (3, 3). :
Put x = 3, y = 3 in the given equation
x + 2y = 9 ⇒ 3 + 2 (3) = 9
⇒ 3 + 6 = 9 ⇒ 9 = 9
∴ LHS = RHS
Hence (3, 3) is another solution to x + 2y = 9
→ How to find different solutions for given linear equation in two variables :
Step 1 : Write the given linear equa-tion. Ex : x + y = P
Step 2 : Put x = some value (your choice)
For example x = 0, then
x + y = P becomes
0 + y = p ⇒ y = p
∴ x = 0, y = p (0, p) is a solution.
Step 3 : Put x = 1, then x + y = p becomes
1 + y = p ⇒ y = p – 1
∴ (0, p – 1) is another solution.
Thus you can repeat the same procedure and get any number of solutions you want.
→ Equations like x + 7 = 10; y + √3 = 8 are examples of linear equations in one variable.
→ If a linear equation has two variables then it is called a linear equation in two variables. Eg.: 3x – 5y = 8; 5x + 7y = 6 ….
→ The general form of a linear equation in two variables x and y is ax + by + c = 0; where a, b, c are real numbers and a, b are not simultaneously zero.
→ Any pair of values of x and y which satisfy ax + by + c = 0 is called the solution of linear equation.
→ An easy way of getting two solutions is put x = 0 and get the corresponding value of y. Similarly put y = 0 and get the value for x.
→ The line obtained by joining all points which are solutions of a linear equation is called graph of linear equation.
→ Equation of a line parallel to X-axis is y = k. (at a distance ‘k’ units)
→ Equation of a line parallel to Y-axis at a distance of k – units is x = k.
→ Equation of X-axis is y = 0 and Y-axis is x = 0.
→ The graph of x = k is a line parallel to Y-axis at a distance of ‘k’ units and passing through the point (k, 0).
→ The graph of y = k is a line parallel to X-axis at a distance of k – units and passing through the point (0, k).