Summary and Quick Revision

Chapter at a Glance

Here is the whole of Pair of Linear Equations on one page — perfect for the night before your exam.

Basics

• A linear equation in two variables: $ax + by + c = 0$ ($a, b$ not both zero).
• A single such equation has infinitely many solutions (a straight line of points).
• A pair has a common solution where the two lines meet.
• Consistent = has a solution (one or infinitely many); Inconsistent = no solution.

Remember: A solution of a pair must satisfy BOTH equations simultaneously.

The Ratio Conditions (Most Important)

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

Condition	Lines	Solutions	Type
$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$	Intersecting	Unique	Consistent
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	Coincident	Infinitely many	Consistent (dependent)
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$	Parallel	None	Inconsistent

Key Point: Write both equations as $ax + by + c = 0$ first, then compare the three ratios. This answers almost all 'how many solutions' and 'find $k$' questions.

The Four Methods

1. Graphical

Draw both lines; the intersection point is the solution. Intersecting → unique, coincident → infinite, parallel → none.

2. Substitution

Isolate one variable from one equation; substitute into the other.

3. Elimination

Make one variable's coefficients equal; add (opposite signs) or subtract (same signs) to cancel it.

4. Cross-multiplication

$$\frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1}$$

Key Point: In substitution/elimination, a false leftover ($0 = 5$) means no solution; a true leftover ($0 = 0$) means infinitely many.

Reducible Equations and Word-Problem Cues

Reducible to linear form

For equations with $\dfrac{1}{x}$ and $\dfrac{1}{y}$, substitute $u = \dfrac{1}{x}$, $v = \dfrac{1}{y}$, solve the linear pair, then convert back ($x = 1/u$, $y = 1/v$).

Word-problem cues

• Two-digit number with digits $x$ (tens), $y$ (units): value $= 10x + y$; reversed $= 10y + x$.
• Boat: downstream speed $= x + y$, upstream $= x - y$.
• Two cars toward each other ⇒ sum of speeds; same direction ⇒ difference.
• 'Fixed + variable' cost ⇒ $x + ny =$ total.

Key Point: Always define variables with units and state the final answer in words.

Last-Minute Tips and Common Traps

• Put equations in $ax + by + c = 0$ form before using ratios or cross-multiplication.
• The sum-of-ratios for 'no solution' needs the $c$-ratio to be different; for 'infinitely many' all three must be equal — always check the third ratio.
• In reducible problems, don't forget to convert $u, v$ back to $x, y$.
• For 'find $k$' questions, write the exact ratio condition for the required case, then solve.
• Verify your solution in both original equations.

Final Word: This is a high-scoring, methodical chapter. Master the ratio table, the four solving methods, and the reducible-equation trick, and you secure most of the marks. All the best!