What this chapter is about
This chapter teaches you how to work with two linear equations that share two unknowns. In earlier classes, you solved single linear equations in one variable. Now you learn to handle situations where two conditions must be satisfied at the same time, leading to two equations that must be solved together.
You will learn what it means for a pair of equations to have exactly one solution, no solution, or infinitely many solutions. The chapter shows you how to find solutions using algebraic methods (substitution and elimination) and how to interpret solutions graphically as the point where two lines meet.
After studying this chapter, you should be able to set up pairs of equations from word problems, choose an efficient method to solve them, and understand what the solution means in the original context.
Key ideas
- A linear equation in two variables has the form ax + by + c = 0, where a, b, c are real numbers and at least one of a or b is non-zero. Its graph is a straight line.
- A pair of linear equations represents two straight lines. The solution of the pair is a point (x, y) that satisfies both equations simultaneously.
- If two lines intersect at exactly one point, the pair has a unique solution; if they are parallel, the pair has no solution; if they coincide (same line), the pair has infinitely many solutions.
- For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare the ratios a₁/a₂, b₁/b₂ and c₁/c₂ to decide the type of solution without solving.
- In substitution method, express one variable in terms of the other from one equation, then substitute into the second equation.
- In elimination method, multiply equations by suitable numbers so that one variable has equal (or opposite) coefficients, then add or subtract to eliminate that variable.
- A consistent pair has at least one solution; an inconsistent pair has no solution.
Formulas and facts to remember
For the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- If a₁/a₂ ≠ b₁/b₂, lines intersect → exactly one solution (consistent).
- If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, lines are parallel → no solution (inconsistent).
- If a₁/a₂ = b₁/b₂ = c₁/c₂, lines coincide → infinitely many solutions (consistent, dependent).
- Substitution: From one equation, get y = (expression in x), put it in the other equation, solve for x, then find y.
- Elimination: Make coefficients of one variable same in magnitude, add or subtract equations, solve for the remaining variable.
Worked examples
Example 1: Solving by substitution
Ravi bought 3 notebooks and 2 pens for ₹85. Meena bought 2 notebooks and 3 pens for ₹80. Find the cost of one notebook and one pen.
Let cost of one notebook = ₹x, cost of one pen = ₹y.
Equation 1: 3x + 2y = 85 Equation 2: 2x + 3y = 80
From Equation 1: 2y = 85 − 3x, so y = (85 − 3x)/2.
Substitute in Equation 2: 2x + 3 × (85 − 3x)/2 = 80 Multiply both sides by 2: 4x + 3(85 − 3x) = 160 4x + 255 − 9x = 160 −5x = 160 − 255 = −95 x = 19
Now y = (85 − 3 × 19)/2 = (85 − 57)/2 = 28/2 = 14.
One notebook costs ₹19 and one pen costs ₹14.
Example 2: Solving by elimination
Solve: 4x + 5y = 23 and 3x + 4y = 18.
Multiply the first equation by 3: 12x + 15y = 69 Multiply the second equation by 4: 12x + 16y = 72
Subtract the first from the second: (12x + 16y) − (12x + 15y) = 72 − 69 y = 3
Put y = 3 in 4x + 5y = 23: 4x + 15 = 23, so 4x = 8, x = 2.
Solution: x = 2, y = 3.
Example 3: Checking consistency using ratios
Determine whether 6x − 4y = 10 and 9x − 6y = 15 have one solution, no solution, or infinitely many solutions.
Write in standard form: 6x − 4y − 10 = 0 and 9x − 6y − 15 = 0.
a₁/a₂ = 6/9 = 2/3 b₁/b₂ = (−4)/(−6) = 2/3 c₁/c₂ = (−10)/(−15) = 2/3
Since a₁/a₂ = b₁/b₂ = c₁/c₂, the two lines coincide. The pair has infinitely many solutions.
Common mistakes
Forgetting to change signs when subtracting entire equations → write out the subtraction term by term before simplifying.
Substituting incorrectly by mixing up which variable was expressed → always replace the exact expression you derived.
Confusing the ratio conditions: thinking equal first two ratios always means no solution → check the third ratio too; equal all three means infinitely many solutions.
Dropping a negative sign when clearing fractions → multiply every term, including constants, on both sides.
Stopping after finding one variable → always substitute back to find the second variable and verify in both original equations.
Quick revision
- Two linear equations in x and y can have one solution, no solution, or infinitely many solutions.
- Compare a₁/a₂ with b₁/b₂ and c₁/c₂ to predict the type of solution without solving.
- Substitution works well when one equation easily gives a variable in terms of the other.
- Elimination works well when coefficients can be matched by simple multiplication.
- Always verify your answer by substituting both values into both original equations.