Linear Equations in Two Variables

Chapter 1 | Class 10

Maharashtra State Board + CBSE Concepts

1. What is a Linear Equation?

An equation which contains two variables and the degree of each term containing a variable is one, is called a linear equation in two variables.

Key Rules:

Must have exactly 2 variables (like \(x\) and \(y\)).
The highest power (index) of variables must be exactly 1.
No terms like \(xy\), \(x^2\), or \(\sqrt{x}\).

2. Standard Form

a\(x\) + b\(y\) + c = 0

Where:

\(a, b, c\) are real numbers.
Important Condition: \(a \neq 0\) and \(b \neq 0\) at the same time.

3. How to check if it's a Linear Equation?

❌ \(3x^2 - 7y = 13\) (Power of x is 2)

💯\(\sqrt{2}x - \sqrt{5}y = 16\) (Wait, this IS linear! The root is on the number, not the variable!)

❌ \(0x + 6y - 3 = 0\) (a is 0, so only 1 variable is left)

❌ \(4xy - 5y - 8 = 0\) (Degree of xy term is 1+1 = 2)

Practice Q1

Identify which of the following are linear equations in 2 variables:

1) \(4m + 3n = 12\)
2) \(\frac{4}{x} + \frac{5}{y} = 4\)
3) \(0.3x + 0y - 36 = 0\)
4) \(4xy - 5y - 8 = 0\)

4. Simultaneous Linear Equations

When we think about two linear equations in two variables at the same time, they are called simultaneous equations.

Equation 1: \(a_1x + b_1y = c_1\)
Equation 2: \(a_2x + b_2y = c_2\)

Solving them means finding the values of \(x\) and \(y\) that satisfy BOTH equations simultaneously.

5. Method 1: Elimination Method

Goal: Eliminate one variable to solve for the other.

Steps to follow:

Step 1: Make the coefficients of one variable (either \(x\) or \(y\)) same in both equations by multiplying.
Step 2: Add or Subtract the equations to eliminate that variable. (Same sign = Subtract, Opposite sign = Add).
Step 3: Solve for the remaining variable.
Step 4: Substitute this value back into any equation to find the first variable.

Practice Q2

Solve the following simultaneous equations using elimination:

\(5x - 3y = 8\)
\(3x + y = 2\)

Practice Q3

Solve the following simultaneous equations:

\(5m - 3n = 19\)
\(m - 6n = -7\)

6. Special Type of Equations

Sometimes the coefficients of \(x\) and \(y\) are interchanged in the two equations.

\(15x + 17y = 21\)
\(17x + 15y = 11\)

The Trick: Do not multiply by huge numbers!

1. Add them to get a simple equation (x + y = c).

2. Subtract them to get another simple equation (x - y = d).

3. Solve the two new simple equations.

Practice Q4

Solve the special type of equations:

\(99x + 101y = 499\)
\(101x + 99y = 501\)

7. Method 2: Substitution Method

Instead of eliminating, we can substitute!

Step 1: Express one variable in terms of the other from one equation. (e.g., \(x = 5 - 2y\))
Step 2: Substitute this expression into the other equation.
Step 3: You now have an equation with only one variable. Solve it!

Practice Q5

Solve using the substitution method:

\(x + 7y = 10\)
\(3x - 2y = 7\)

8. Graphical Method

A linear equation in two variables represents a straight line on a graph.

To draw a line, we need at least 3 ordered pairs \((x, y)\).

Why 3? Two points are enough to draw a line, but the third point acts as a check! If they don't form a straight line, you made a calculation mistake.

9. How to Make a Graph Table

Example: \(x + y = 4\)

x	0	2	4
y	4	2	0
(x,y)	(0,4)	(2,2)	(4,0)

Choose simple values for \(x\) to avoid fractions if possible!

Practice Q6

Complete the table to draw the graph of the equation:

\(3x - y = 2\)

10. Solving Equations Graphically

When you graph two simultaneous equations, you get two straight lines.

The point of intersection of the two lines is the solution \((x, y)\) of the simultaneous equations!

Practice Q7

Solve the following equations graphically:

\(x + y = 5\)
\(x - y = 3\)

(Draw the tables and sketch the conceptual intersection point on the board)

11. Determinants

A determinant is a number associated with an arrangement of numbers in rows and columns.

\( \begin{vmatrix} a1 & b1 \\ a2 & b2 \end{vmatrix} \) = \( (a1 \times b2) - (b1 \times a2) \)

We cross-multiply and subtract!

Practice Q8

Find the values of the following determinants:

1) \( \begin{vmatrix} 5 & 3 \\ -7 & 0 \end{vmatrix} \)

2) \( \begin{vmatrix} \frac{7}{3} & \frac{5}{3} \\ \frac{3}{2} & \frac{1}{2} \end{vmatrix} \)

12. Determinant Method (Cramer's Rule)

Gabriel Cramer created a simple formula using determinants to solve simultaneous equations.

Given: \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\)

\( D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \)

\( D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \)

\( D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \)

Formulas: \( x = \frac{D_x}{D} \) and \( y = \frac{D_y}{D} \)

13. Cramer's Rule Memory Trick

Remember the columns: 1 (x coeff), 2 (y coeff), 3 (constants).

For D: Use columns 1 and 2.
For D\(_x\): Hide column 1, use 3 and 2. (Constants replace x).
For D\(_y\): Hide column 2, use 1 and 3. (Constants replace y).

Practice Q9

Solve using Cramer's Rule:

\(3x - 4y = 10\)
\(4x + 3y = 5\)

Practice Q10

Solve using Cramer's Rule (Rearrange first!):

\(4m + 6n = 54\)
\(3m = 28 - 2n\)

14. Equations Reducible to Linear Form

Some equations aren't linear, but we can disguise them to make them linear!

\( \frac{4}{x} + \frac{5}{y} = 7 \)

Substitution Trick:
Let \( \frac{1}{x} = m \) and \( \frac{1}{y} = n \)

The equation becomes: \( 4m + 5n = 7 \) (Now it's linear!)

Practice Q11

Solve the following simultaneous equations:

\( \frac{2}{x} - \frac{3}{y} = 15 \)

\( \frac{8}{x} + \frac{5}{y} = 77 \)

15. Nature of Solutions (Concept Booster)

Without even graphing, we can predict what the graph will look like by comparing the ratios of coefficients: \(\frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2}\).

Condition 1: Intersecting Lines
If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
Result: Exactly ONE unique solution.

16. Nature of Solutions: Parallel Lines

Condition 2: Parallel Lines
If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
Result: NO solution (The lines never meet).

Example:
\(2x + 3y = 5\)
\(4x + 6y = 15\)
(Ratios: 1/2 = 1/2 ≠ 1/3)

17. Nature of Solutions: Coincident Lines

Condition 3: Coincident (Overlapping) Lines
If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
Result: Infinitely MANY solutions.

Example:
\(x + 2y = 4\)
\(2x + 4y = 8\)
(All ratios are exactly 1/2)

Practice Q12

Predict whether the lines representing the following pair of linear equations intersect at a point, are parallel, or coincident.

\(5x - 4y + 8 = 0\)
\(7x + 6y - 9 = 0\)

18. Application: Word Problems

How to approach Word Problems:

Read carefully and identify the two unknown quantities.
Assign variables (let them be \(x\) and \(y\)).
Form Equation 1 from the first condition given in the problem.
Form Equation 2 from the second condition.
Solve using any method!

Practice Q13 (Age Problem)

The sum of father's age and twice the age of his son is 70. If we double the age of the father and add it to the age of his son the sum is 95. Find their present ages.

Practice Q14 (Fraction Problem)

The denominator of a fraction is 4 more than twice its numerator. Denominator becomes 12 times the numerator, if both the numerator and the denominator are reduced by 6. Find the fraction.

Practice Q15 (Two-Digit Number)

The sum of a two digit number and the number obtained by interchanging the digits is 143. If the digit at the tens place is 3 more than the digit at the units place, find the original number.

Practice Q16 (Speed/Distance)

A boat travels 16 km upstream and 24 km downstream in 6 hours. The same boat travels 36 km upstream and 48 km downstream in 13 hours. Find the speed of water current and speed of boat in still water.

Chapter Complete!

You've mastered Linear Equations!

Use the smartboard canvas to clear up any final doubts.