PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
Frequently Asked Questions
An equation that can be written in the form \(ax + by + c = 0\), where \(a, b, c\) are real numbers and \(a\) and \(b\) are not both zero.
Two linear equations involving the same variables \(x\) and \(y\) that are solved together to find common solutions.
\(a x + b y + c = 0\), where \(a\), \(b\), \(c\) are constants.
A pair of values \((x, y)\) that satisfies both equations simultaneously.
Two straight lines on a coordinate plane.
(i) One solution, (ii) No solution, (iii) Infinitely many solutions.
When the two lines intersect at exactly one point.
\(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
When the lines are parallel and never intersect.
\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
When both equations represent the same line (coincident lines).
\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
Plotting both equations as lines and finding their point of intersection.
The common solution of both equations.
A pair of equations with at least one solution (unique or infinite).
A pair of equations with no solution.
Equations representing the same line (infinitely many solutions).
Equations representing two different lines (unique solution).
Solving one equation for one variable and substituting it into the other.
Eliminating one variable by adding/subtracting appropriately modified equations.
A formulaic method using \(\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}\).
When the denominator \(a_1b_2 - a_2b_1 \neq 0\).
A straight line representing all solutions of the equation.
The ratio of change in \(y\) to change in \(x\).
Check if the lines intersect or coincide.
\(2x + 3y = 6,; x - y = 1\).
Adding gives \(2x = 12\), so \(x = 6,; y = 4\).
From 2nd eq.: \(y = x - 1\). Substitute: \(2x + x - 1 = 7 \Rightarrow x = 8/3,; y = 5/3.\)
Profit–loss, age problems, mixture problems, speed-distance-time, cost calculations, geometry.
To determine the nature of lines and number of solutions.
Two equations solved together to find the same pair of variables.
Both lines lie on top of each other.
Rearrange terms to match \(ax + by + c = 0\).
It helps check the determinant and decide the nature of solutions.
\(D = a_1b_2 - a_2b_1\). It indicates uniqueness of solution.
Either no solution or infinitely many solutions.
One unique solution exists.
\(2x + 3y = 5\) and \(4x + 6y = 10\).
\(x + y = 5\) and \(x - y = 1\).
\(2x + 4y = 8\) and \(x + 2y = 4\).
Create equal coefficients for one variable and eliminate it.
Replace one variable with an equivalent expression.
Human drawing and scale errors may lead to approximations.
When coefficients are easily manageable to eliminate a variable.
When one variable is already isolated or easy to isolate.
Multiplying equations by constants and adding them to eliminate a variable.
At least two solution points.
Because the relationship between \(x\) and \(y\) is constant and proportional.
A pair \((x, y)\) that represents a point on a coordinate plane.
Nature of solutions, solving methods, word problems, graphing, ratio comparison.
Substitute into both equations to check if they hold true.
Simplifying equations by dividing by common factors.
Calculating cost, profit, discount, selling price, mixtures.
When their slopes are different.
Because they never meet and rise at the same rate.
Duplicate measurements or equal ratios in daily problems.
Provides quick calculation steps for board exams.
It shifts the line up/down or left/right on a graph.
No — they can have only 1, 0, or infinite solutions.
Forms the base for linear algebra, matrices, determinants, and advanced maths.
