Linear Equations in Two Variables Maths Notes

Linear Equations in Two Variables Maths Notes

Linear Equation in Two Variables

An equation in two variables with the degree of the variables 1, is called a linear equation in two variables.

e.g. (i) 5x + 3y = 9 (ii) 3x – 5y = 2 are linear equations in two variables x and y. The degree of 1 the variables x and y is 1.

Note : (i) \(\frac{2}{x}\) + \(\frac{5}{x}\) = 7 is not a linear equation in two variables, because the degree of the variables x and y is -1.

But by proper supposition, it can be converted into linear equation in two variables.

Substituting m for \(\frac{1}{x}\) and n for \(\frac{1}{y}\), the given equation becomes, 2m + 5n = 7, which is a linear equation.

(ii) 3x² + 5y² = 10 is not a linear equation in two variables, because the degree of the variables x and y is 2. Such equation cannot be converted into linear equation.

Through this activity, we can recognize linear equations in two variables.

Simultaneous Linear Equations

Two linear equations involving two variables taken together form a system .of simultaneous linear equations.

e.g.

• 3x – 4y = 2; 5x + 3y = 13
• 2p + 3q = 12; 4p-5q = 2.

Methods of Solving Simultaneous Equations :

• The method of elimination by equating the coefficients
• The method of elimination by substitution
• By graphical method
• By Cramer’s rule

Equations Reducible to a Pair of Linear Equations in Two Variables.
While solving an equation, always keep the following points in mind :
The solution of a linear equation is not affected when

• the same number is added to (or subtracted from) both the sides of an equation.
• multiplying or dividing both the sides of an equation by the same non-zero number.

Solving simultaneous equations by the method of elimination.
Solve:
3x + 2y = 29; 5x – y = 18
Solution:
3x + 2y = 29 ……….. (1)
5x – y = 18 …………. (2)
Multiplying equation (2) by 2,

Graph of a Linear Equation in Two Variables

The graph of a linear equation in two variables is a line.
The ordered pair which satisfies the equation is a solution of that equation.
The ordered pair represents a point on that line.

Graphical Method

Steps to follow for drawing a Graph of Linear Equation in two variables :

1. Choose at least three convenient values of x and find the corresponding values of y satisfying the given equation. Write the values of x and y in tabular form.
2. Draw X-axis, Y-axis on graph paper and plot the j points.
3. See that all the points lie on a line.
4. The coordinates of the point of intersection of the two lines is the solution of the given equations.

Activity 1

Solve the simultaneous equations by graphical method:
Complete the following tables to get ordered pairs : x – y = 1

The coordinates of the point of intersection are (-1, -2)
(x, y) = (-1, 2) is the solution.

Activity 2

Solve the above equations by the method of elimination. Check your solution with the solution obtained by graphical method.
x – y = 1 … (1)
5x – 3y = 1 … (2)
Multiplying equation (1) by 3,
3x – 3y = 3 … (3)
Subtracting equation (3) from equation (2),

Substituting x = – 1 in equation (1),
– 1 – y = 1 ∴ – y = 1 + 1 ∴ – y = 2 ∴ y = – 2.
( – 1, – 2) is the solution of the given equations.
The solution in both the methods is the same.

Let’s think

The following table contains the values of x and y coordinates for ordered pairs to draw the graph of 5x – 3y = 1.

1. Is it easy to plot these points?
2. Which precaution is to be taken to find ordered pairs so that plotting of points become easy?

Answer:

1. It is not easy to plot these points except (\(\frac{1}{5}\), 0). The others have 3 in the denominators. These numbers are non-terminating recurring decimals. Hence, their values are to be taken approximately. The positions of the points will not be accurate.
2. The values of x and y should be selected in such a way that the coordinates are integers and not fractional numbers.

Determinant: 

is a determinant of four numbers a, b, c, d.
(a, b), (c, d) are rows and \(\left(\frac{a}{c}\right)\), \(\left(\frac{b}{d}\right)\) are columns.

The degree of the determinant is 2, because there are 2 elements in each column and 2 elements in each row. The determinant represent a number which is (ad – bc).

ad – bc is the value of the determinant
.
Determinants are usually represented with capital letters A, B, C, etc.

Determinant Method (Cramer’s Rule) 

Determinant method of solving simultaneous equations was first given by a Swiss mathematician Gabriel Cramer. So, it is also known as Cramer’s rule.
To use Cramer’s rule, the equations are written as a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
a₁x + b₁y = c₁ …… (1)
a₂x + b₂y = c₁ ………. (2)
Here, x and y are variables, a₁, b₁, c₁ and a₂, b₂, c₂ are real numbers, a₁b₂ – a₂b₁ ≠ 0
Now let us solve these equations.
Multiplying equation (1) by b₂,
a₁b₂x + b₁b₂y = c₁b₂ … (3)
Multiplying equation (2) by b₁,
a₂b₁X + b₂b₁y = c₂b₁ … (4)
Subtracting equation (4) from (3),

To remember and write the expressions
c₁b₂ – c₂b₁, a₁b₂ – a₂b₁, a₁c₂ – a₂c₁ we use the determinants.
Now, a₁x + b₁y = c₁ and a₂x + b₂y = c₂
We can write 3 columns.
\(\left(\frac{a_{1}}{a_{2}}\right),\left(\frac{b_{1}}{b_{2}}\right),\left(\frac{c_{1}}{c_{2}}\right)\)
The values x, y in equations (5), (6) are written using determinants as follows :

Steps to solve simultaneous equations by Cramer’s rule.

1. Write the given equations in the form ax + by = c.

x = \(\frac{\mathrm{D}_{x}}{\mathrm{D}}\) and y = \(\frac{\mathrm{D}_{y}}{\mathrm{D}}\)

Activity 1
To solve the simultaneous equations by determinant method, fill in the blanks :
y + 2x – 19 = 0; 2x – 3y + 3 = 0.
Solution:
Write the given equations in the form
ax + by = c
2x + y = 19
2x – 3y = – 3

By Cramer’s Rule –

Activity 2

Complete the following activity :

Let’s think 

What is the nature of the solution if D = 0?

∴ the two simultaneous equations have either no solution or infinite solutions. It means the two simultaneous equations do not have a unique solution.

What can you say about lines if common solution is not possible?
If the common solution about the lines is not possible, then the lines will either coincide or the lines are parallel to each other.

Let’s think

In the above table the equations are not linear. Can you convert the equations into linear equations?

We can create new variables making a proper change in the given variables. Substituting the new variables in the given non-linear equations, we can convert them in linear equations.

Study the following example :
\(\frac{4}{x}+\frac{5}{y}\) = 7; \(\frac{3}{x}+\frac{4}{y}\) = 5;
Solution:

Substituting a for \(\frac{1}{x}\) and b for \(\frac{1}{y}\)
4a + 5b = 7 ………… (3)
3a + 4b = 5 …………. (4)
These are linear equations in variables a and b.
Multiplying both the sides of equation (3) by 4 and equation (4) by 5,

Substituting a = 3 in equation (3),
4(3) + 5b = 7
∴ 12 + 5b = 7
∴ 5b = 7 – 12 ∴ 5b = -5 ∴ b = – 1.
Now, a = \(\frac{1}{x}\) = 3 ∴ x = \(\frac{1}{3}\); b = \(\frac{1}{y}\) = – 1 ∴ y = -1
Ans. (x, y) = (\(\frac{1}{3}\), – 1) is the solution of the given simultaneous equations.

Let’s think 

If you solve the above equations [Ex. (2)] by graphical method and by Cramer’s rule will you get the same answers?
⇒ Student should solve these equations by the two methods.
See that the solution will be the same.

Activity:

To solve given equations fill the boxes below suitably

Application of Simultaneous Equations

Use of simultaneous equations is common in Science and Mathematics. The simultaneous equations are used to solve word problems.
To solve the problem, apply the following steps :

1. Read the problem carefully. Identify the unknown quantities about which some information is supplied. Denote these quantities by variables like x, y, a, b, etc.
2. Concentrate on each phrase of the problem one by one. Write the equation in terms of the variables which these phrases suggest.
3. Solve the equations and find the values of the variables and write the answer.

Activity

There are some instructions given below. Frame the equations from the informations and write them in the blank boxes shown below by arrows.

Maths Notes