Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 3: Pair of Linear Equations in Two Variables — notes
The chapter’s 7 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. What a pair of linear equations looks like
Okay, so let's start with something you already know from Class IX.
- • Linear equation: ax + by + c = 0
- • Pair: a1x + b1y + c1 = 0, a2x + b2y + c2 = 0
- • Example: 2x + 3y = 5
- • x = 1, y = 1 → LHS = 5 = RHS (solution)
- • x = 1, y = 7 → LHS = 23 ≠ RHS (not a solution)
- • Akhila: y = (1/2)x and 3x + 4y = 20
2. Three ways two lines can behave, and solving by graph
Okay, so think about this.
- • Intersecting lines → unique solution (consistent)
- • Parallel lines → no solution (inconsistent)
- • Coincident lines → infinite solutions (dependent, consistent)
- • Akhila's lines meet at (4,2): 4 rides, 2 Hoopla
- • Pencils/erasers: lines coincide, many solutions
- • Rails: lines parallel, no solution
3. Comparing coefficients: a1/a2, b1/b2, c1/c2
You, there is a quick way to tell, just from the numbers, what kind of pair you have, without drawing anything.
- • a1/a2 ≠ b1/b2 → intersecting, unique solution
- • a1/a2 = b1/b2 = c1/c2 → coincident, infinite solutions
- • a1/a2 = b1/b2 ≠ c1/c2 → parallel, no solution
- • Example: x − 2y = 0, 3x + 4y − 20 = 0
- • 1/3 ≠ -2/4 → lines intersect
4. Substitution method
Okay, so you know the graphical method can be tricky when the solution isn't a whole number.
- • Step 1: write one variable in terms of the other
- • Step 2: substitute into the other equation
- • Step 3: solve for one variable, then find the other
- • Example: 7x − 15y = 2, x + 2y = 3
- • x = 3 − 2y → 7(3−2y) − 15y = 2 → y = 19/29
- • x = 3 − 2(19/29) = 49/29
5. Elimination method
Okay, so there's another way to solve these pairs — it's called the elimination method, and it's often faster.
- • Step 1: make one variable's coefficient equal
- • Step 2: add or subtract to eliminate it
- • Step 3: solve the remaining equation
- • Step 4: substitute back for the other variable
- • Example: 9x − 4y = 2000, 7x − 3y = 2000
- • x = 2000, y = 4000 → incomes ₹18,000 and ₹14,000
6. Cross-multiplication method
Okay, so there's one more algebraic method — cross-multiplication.
- • x/(b1c2 − b2c1) = y/(c1a2 − c2a1) = 1/(a1b2 − a2b1)
- • Needs a1b2 − a2b1 ≠ 0
- • Example: 2x+3y−46 = 0, 3x+5y−74 = 0
- • x/8 = y/10 = 1/1
- • x = 8, y = 10 (fares ₹8 and ₹10)
7. Equations reducible to a pair of linear equations
Okay, so some equations don't look linear at first.
- • 2/x + 3/y = 13, 5/x − 4/y = -2
- • Put p = 1/x, q = 1/y
- • 2p + 3q = 13, 5p − 4q = -2
- • Solve: p = 2, q = 3
- • So x = 1/2, y = 1/3
Summary
Okay, so let's look back at everything you've learned in this chapter. A linear equation in two variables is ax + by + c = 0. Two such equations together give a pair of linear equations. These are two straight lines, and they can intersect, or be parallel, or lie on top of each other — coincident. That gives a unique solution, no solution, or infinitely many solutions. You learned to tell which case it is just by comparing a1/a2, b1/b2, c1/c2 — no graph needed for that. Then came three ways to find the exact solution. In substitution, you write one variable in terms of the other. In elimination, you make the coefficients equal, then add or subtract. In cross-multiplication, you use the formula with a1, b1, c1, a2, b2, c2. And some equations aren't linear at all — the ones with 1/x and 1/y. You saw how a substitution, like p = 1/x and q = 1/y, turns them into a pair of linear equations. After that, they're solved the same way as before. So, out of all these methods, which one did you find the quickest to use?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 3. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Pair of Linear Equations in Two Variables" notes as a PDF?
- Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
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