Two unknowns, two equations
One fact is never enough
'My father is 30 years older than me' fits many pairs of ages.
↳ Two unknowns generally need two equations.
📚 Chapter Learning Kit
Pair of Linear Equations in Two Variables · Class 10 Mathematics
English 4 topics 220 leveled MCQs 80 flashcards 3 games Free
Shared by a Veda teacher · Generated with Veda AI · Sep 2026
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💡 Key Idea
📖 Definition
⚙️ Process
💡 Key Idea
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➗ Formula
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Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. Here's a taste from this kit; the app has the full deck.
General form of a pair
The shape to recognise
↳ A solution must satisfy BOTH equations.
Turning words into equations
Name the letters first
↳ The 'Let' line carries its own mark.
Decide before you draw
The coefficients already know
Put both equations as $a_1x + b_1y + c_1 = 0$ and compare $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$.
↳ No graph paper needed.
One solution
Different at the first step
The constants do not matter in this case.
↳ Unequal a-to-b ratios mean the lines must cross.
Infinitely many solutions
All three agree
↳ One equation is a multiple of the other.
📖 Smart notes
What you'll study, topic by topic
1
Making a Pair of Equations and Solving It by Graph
Two unknowns need two equations. Turn the words of a problem into a pair of linear equations, draw both lines from two points each, and the point where they cross is the solution — the one pair of values that satisfies b…
- A LINEAR EQUATION IN TWO VARIABLES has the form $ax + by + c = 0$, where a and b are not both zero.
- A PAIR of such equations is written $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
- A SOLUTION of the pair is a pair of values (x, y) that satisfies BOTH equations.
~40 min · full explanation, examples & memory tricks in the app
2
How Many Solutions? The Ratio Test
You can tell what a pair of lines will do before drawing anything. Compare a₁/a₂ with b₁/b₂ and c₁/c₂: different ratios mean one solution, all three equal means infinitely many, and the first two equal with the third dif…
- Put both equations in the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ before comparing anything.
- $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$ → the lines INTERSECT → exactly ONE solution (consistent).
- $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ → the lines COINCIDE → INFINITELY MANY solutions (dependent, consistent).
~40 min · full explanation, examples & memory tricks in the app
3
The Substitution Method
Make one variable the subject in one equation, put that expression into the other, and a pair in two unknowns becomes a single equation in one. Substitution is the method of choice whenever a variable already stands alon…
- STEP 1: from one equation, write one variable in terms of the other, e.g. $x = 3 - 2y$.
- STEP 2: substitute that expression into the OTHER equation — never back into the same one.
- STEP 3: solve the resulting one-variable equation, then substitute back to get the second value.
~40 min · full explanation, examples & memory tricks in the app
4
The Elimination Method and Word Problems
Multiply the equations so that one variable has the same coefficient in both, then add or subtract to wipe it out. Elimination is the workhorse when no coefficient is 1, and it is the method most word problems are solved…
- STEP 1: multiply one or both equations so that one variable has the SAME numerical coefficient in both.
- STEP 2: if those coefficients have the same sign, SUBTRACT; if opposite signs, ADD.
- STEP 3: solve the one-variable equation, then substitute back into the simpler original equation.
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
220 questions laddered from warm-up to topper-level, each with an explanation. A taste:
A linear equation in two variables has the form
Beginner
A $ax^2 + by + c = 0$
B $\frac{a}{x} + by = c$
C $ax + by + c = 0$ with a, b not both zero
D $ax + b = 0$
Show answer & explanation
$ax + by + c = 0$ with a, b not both zero
Both variables appear to the first power, and at least one of a, b is non-zero.
A solution of a pair of linear equations is a pair (x, y) that
Beginner
A satisfies at least one equation
B satisfies both equations
C makes both sides zero
D lies on the x-axis
Show answer & explanation
satisfies both equations
It must work in both equations at the same time.
The graph of a linear equation in two variables is
Beginner
A a parabola
B a straight line
C a pair of lines
D a circle
Show answer & explanation
a straight line
That is why they are called linear.
If two lines cross at exactly one point, the pair of equations has
Beginner
A no solution
B infinitely many solutions
C exactly one solution
D two solutions
Show answer & explanation
exactly one solution
The crossing point is the only pair satisfying both.
🃏 Flashcards
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80 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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Word Match
True False
Fill Blank
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