Notes: Pair of Linear Equations in Two Variables
These notes are based directly on the uploaded Class X Mathematics chapter. The focus is on concepts, methods, formulas, and key results, without exercise questions.
1. Pair of Linear Equations in Two Variables
A pair of linear equations in two variables can be written as a1x+b1y+c1=0
and a2x+b2y+c2=0
where x and y are variables.
The solution of the pair is the ordered pair (x,y) that satisfies both equations simultaneously.
2. Graphical Meaning
Each linear equation in two variables represents a straight line. Therefore, solving a pair of linear equations means finding the common point(s) of the two lines.
There are three possibilities:
A. Intersecting Lines
- The two lines meet at exactly one point.
- There is one unique solution.
- The pair is consistent.
B. Parallel Lines
- The two lines never meet.
- There is no solution.
- The pair is inconsistent.
C. Coincident Lines
- Both equations represent the same line.
- Every point on the line is a common solution.
- There are infinitely many solutions.
- The pair is dependent and consistent.
Quick picture to remember
Intersecting → 1 solution
Parallel → 0 solutions
Coincident → infinitely many solutions
3. Algebraic Conditions for the Nature of Solutions
For a1x+b1y+c1=0
and a2x+b2y+c2=0,
compare the ratios of corresponding coefficients.
Unique solution / Consistent
a2a1=b2b1
The lines intersect at one point.
No solution / Inconsistent
a2a1=b2b1=c2c1
The lines are parallel.
Infinitely many solutions / Dependent and consistent
a2a1=b2b1=c2c1
The lines are coincident.
4. Graphical Method
The graphical method involves drawing the two lines represented by the equations.
Steps
- Take the first equation and find at least two points satisfying it.
- Plot those points and draw the first line.
- Find two points satisfying the second equation.
- Plot them and draw the second line.
- Observe the intersection of the lines.
- The coordinates of the common point give the solution.
For example, if the lines meet at (6,0), then x=6,y=0
is the solution.
Limitation of the graphical method
Graphical solutions can be inconvenient when the coordinates are fractional or decimal values, because accurately reading such points from a graph can be difficult. Algebraic methods are therefore often more convenient.
5. Substitution Method
The substitution method solves one equation for one variable and substitutes that expression into the other equation.
Main procedure
Suppose we have two equations.
Step 1: Express one variable in terms of the other
From either equation, obtain something such as x=f(y)
or y=f(x).
Choose the equation that makes this easiest.
Step 2: Substitute
Put this expression into the other equation. The result will contain only one variable.
Solve it to obtain its value.
Step 3: Find the other variable
Substitute the value obtained in Step 2 into the expression from Step 1.
Step 4: Verify
Substitute both values into the original equations to check that they satisfy both equations.
Example from the chapter
Given 7x−15y=2 x+2y=3
From the second equation, x=3−2y
Substituting in the first: 7(3−2y)−15y=2 21−14y−15y=2 −29y=−19
so y=2919.
Then x=3−2(2919)=2949.
Therefore, x=2949,y=2919.
6. What Happens in Substitution for Special Cases?
During substitution, the resulting equation may contain no variable.
True statement
For example, 18=18
is always true.
This indicates that the equations have infinitely many solutions.
False statement
For example, −4=0
is impossible.
This indicates that the equations have no solution and are inconsistent.
7. Elimination Method
The elimination method removes one of the variables by making its coefficients equal and then adding or subtracting the equations.
Steps
Step 1: Multiply one or both equations by suitable non-zero numbers so that the coefficients of one variable become numerically equal.
Step 2: Add or subtract the equations to eliminate that variable.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute this value into either original equation to find the other variable.
Example structure
Suppose 9x−4y=2000 7x−3y=2000.
To eliminate y, multiply the first equation by 3 and the second by 4: 27x−12y=6000 28x−12y=8000.
Subtracting, x=2000.
Substitution then gives y=4000.
The original quantities represented by x and y can then be obtained from these values.
8. Special Results in the Elimination Method
After eliminating a variable:
A. Equation in one variable
Example: x=5
A unique value is obtained, so the pair has a unique solution.
B. True statement
Example: 0=0
The equations have infinitely many solutions.
C. False statement
Example: 0=9
The equations have no solution.
9. Important Idea: Converting Real-Life Situations into Equations
Many word problems can be represented using two variables.
The general approach is:
- Choose variables for the unknown quantities.
- Translate each given condition into an equation.
- Obtain a pair of linear equations.
- Solve the equations using a suitable method.
- Interpret the values in terms of the original situation.
- Verify that the result satisfies the conditions.
The chapter applies this idea to situations involving ages, costs, numbers, digits, incomes and expenditures, travel charges, purchases, and other quantities.
10. Representing a Two-Digit Number
This is an important application of linear equations.
If the tens digit is x and the units digit is y, then the number is 10x+y
When the digits are reversed, the number becomes 10y+x.
For example, if the digits are 5 and 6: 56=10(5)+6
and the reversed number is 65=10(6)+5.
This representation allows conditions involving a two-digit number and its reversed form to be converted into linear equations.
11. Choosing Between Methods
Graphical Method
Best for:
- understanding the geometric meaning;
- seeing whether lines intersect, coincide or are parallel;
- obtaining a visual interpretation.
Substitution Method
Useful when:
- one equation can easily be rearranged for one variable;
- a variable already has coefficient 1 or −1.
Elimination Method
Often convenient when:
- coefficients can easily be made equal;
- adding or subtracting the equations can remove one variable directly.
The chapter notes that substitution, elimination and graphical methods can sometimes all be used for the same problem; the convenient method depends on the form of the equations.
12. Consistency at a Glance
| Graphical form | Algebraic condition | Number of solutions |
|---|---|---|
| Intersecting lines | a2a1=b2b1 | 1 |
| Parallel lines | a2a1=b2b1=c2c1 | 0 |
| Coincident lines | a2a1=b2b1=c2c1 | Infinitely many |
The chapter classifies intersecting cases as consistent, parallel cases as inconsistent, and coincident cases as dependent and consistent.
13. Chapter Takeaways
- A pair of linear equations in two variables represents two straight lines.
- Solving the pair means finding their common solution(s).
- Two lines can be intersecting, parallel or coincident.
- Intersecting lines → one solution.
- Parallel lines → no solution.
- Coincident lines → infinitely many solutions.
- The graphical method gives the solution through the common point of the lines.
- The two main algebraic methods are substitution and elimination.
- In substitution, express one variable in terms of the other and substitute.
- In elimination, make the coefficients of one variable equal and eliminate it.
- A true variable-free statement indicates infinitely many solutions.
- A false variable-free statement indicates no solution.
- Real-life problems can be converted into pairs of linear equations by defining suitable variables.