01 · Explore
Introduction to Linear Equations in Two Variables
A linear equation in two variables represents a relationship between two unknown quantities where the highest power of each variable is one.
A pair of linear equations in two variables, x and y, is generally written as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Here, a₁, b₁, c₁, a₂, b₂, and c₂ are real numbers such that (a₁² + b₁²) ≠ 0 and (a₂² + b₂²) ≠ 0. These equations are called 'linear' because their graphs on a Cartesian plane are straight lines.
Consider a real-world scenario: Akhila spends 20 rupees at a fair on Giant Wheel rides (x) and Hoopla games (y). If a ride costs 3 rupees and Hoopla costs 4 rupees, the total cost is 3x + 4y = 20. If she played Hoopla half as many times as she rode the wheel, we get y = (1/2)x. Together, these form a pair of linear equations.
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What is the general form of a pair of linear equations in two variables?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 1, 2, 3.
02 · Explore
Graphical Method and Consistency
The solution to a pair of linear equations corresponds to the point where their two lines meet on a graph.
When we plot two linear equations on a graph, three geometric possibilities exist: the lines intersect at one point, they are parallel, or they coincide (overlap completely). Each case tells us about the number of solutions the system has.
A system is 'consistent' if it has at least one solution. If the lines intersect at a single point, there is a unique solution. If the lines coincide, there are infinitely many solutions, and the system is called 'dependent and consistent'. If the lines are parallel, they never meet; thus, there is no solution, and the system is 'inconsistent'.
| Lines | What the graph means |
|---|---|
| Intersecting Lines | The lines cross at exactly one point (x, y), representing a unique solution. |
| Coincident Lines | The lines lie on top of each other, representing infinitely many solutions. |
| Parallel Lines | The lines never meet, meaning the system has no solution. |
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What do we call a pair of linear equations that has no solution?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 2, 3.
03 · Explore
Comparing Coefficients for Solutions
We can predict the nature of the solutions without drawing a graph by comparing the ratios of the coefficients.
For the equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, we compare the ratios a₁/a₂, b₁/b₂, and c₁/c₂. This comparison provides a shortcut to determine if the lines intersect, are parallel, or coincide.
If a₁/a₂ ≠ b₁/b₂, the lines intersect and there is a unique solution. If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are coincident and there are infinitely many solutions. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and there is no solution.
| Ratio Comparison | Graphical Representation | Algebraic Interpretation |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting lines | Exactly one (unique) solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident lines | Infinitely many solutions |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel lines | No solution |
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If a₁/a₂ = 1/2, b₁/b₂ = 1/2, and c₁/c₂ = 5/6, how many solutions does the system have?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 3, 14.
04 · Explore
The Substitution Method
The substitution method involves expressing one variable in terms of the other to reduce the system to a single-variable equation.
This algebraic method is particularly useful when one of the variables has a coefficient of 1 or -1. By isolating that variable, we can 'substitute' its expression into the second equation. This eliminates one variable, allowing us to solve for the remaining one.
Once the value of the first variable is found, we plug it back into our isolation equation to find the second variable. If during substitution the variables cancel out and leave a true statement (like 18 = 18), the system has infinite solutions. If it leaves a false statement (like -4 = 0), the system has no solution.
Solving by Substitution
7x - 15y = 2 (Eq 1); x + 2y = 3 (Eq 2)
Step 1: From Eq 2, isolate x: x = 3 - 2y. Step 2: Substitute this into Eq 1: 7(3 - 2y) - 15y = 2. Step 3: Expand: 21 - 14y - 15y = 2. Step 4: Simplify: 21 - 29y = 2, so -29y = -19, which means y = 19/29. Step 5: Find x: x = 3 - 2(19/29) = (87 - 38)/29 = 49/29. Check: 7(49/29) − 15(19/29) = 58/29 = 2, and 49/29 + 2(19/29) = 87/29 = 3. Both original equations are satisfied.
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When using substitution, what does it mean if you get a result like 0 = 0?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 7, 8, 9.