Pair of Linear Equations in Two Variables — Class 10 CBSE Mathematics revision notes
A 4-page visual revision sheet for Pair of Linear Equations in Two Variables, written for Class 10 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Pair of Linear Equations in Two Variables in one look
two lines on one grid, and the three things they can do
The standard form of a pair
- first equation
a_1x+b_1y+c_1=0 - second equation
a_2x+b_2y+c_2=0 - neither may vanish completely
a_1^2+b_1^2≠0, a_2^2+b_2^2≠0 - a solution is a point of the plane
(x,y)
What the ratios settle
- a and b ratios differ — lines intersect: one solution, consistent
- a and b ratios equal, c ratio differs — lines parallel: no solution, inconsistent
- all three ratios equal — lines coincide: infinitely many, consistent and dependent
- cross multiplication's own denominator is 0 — lines parallel or coincident, formula gives nothing
Drawing the two lines
- two points fix a line: find two and rule it, no table of five needed
- the quickest two are the intercepts: set x=0 for one point, then y=0 for the other
- for 2x+3y=12 that gives (0, 4) and (6, 0), and the line is drawn
- keep both coordinates whole numbers, or reading off the grid stops being exact
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Pair of Linear Equations in Two Variables sheet covers
- Introduction
- The standard form of a pair
- first equation
- second equation
- neither may vanish completely
- a solution is a point of the plane
- Graphical Method of Solution
- Consistency and Number of Solutions
- What the ratios settle
- Reading the lines off the grid
- Drawing the two lines
- Testing a point against the pair
- does (2, 3) satisfy 2x+3y=13
- and the second line x-y=-1
- both hold, so the solution is
- while (5, 1) only manages the first
- One equation, and then two
- Algebraic Methods
- Substitution Method
- Elimination Method
- The ratio test on three real pairs
- 2x+3y-5=0 with 4x+6y-15=0
- 2x+3y-5=0 with 4x+6y-10=0
- 2x+3y-13=0 with x-y+1=0
- Substitution vs elimination
- Substitution
- Elimination
- Choosing k so the pair behaves
- for kx+2y-5=0 with 3x+y-1=0
- one solution needs the first two apart
- no solution needs them equal, c apart
- so k=6 is the single value that fails
- Cross Multiplication Method
- Cross multiplication
- with both in a x + b y + c = 0 form
- on 2x+3y-13=0 and x-y+1=0
- next term
- last term
- so the point is
- Equations reducible to a linear pair
- given
- swap in
- the pair turns linear
- solve, then undo the swap
- Symbols
- Turning words into a pair
- A two digit number, worked through
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
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