Which of these is a solution of \(2x + 3y = 12\)?
- (a)\((4, 1)\)
- (b)\((2, 3)\)
- (c)\((3, 2)\)
- (d)\((1, 4)\)
Linear Equations in Two Variables Class 9: notes and important questions
Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 17 questions
- 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
- About 8 hours to master
Algebra unit: 20 of 80 theory marks (Introduction to Polynomials, Sequences and Progressions, Exploring Algebraic Identities, Linear Equations in Two Variables).
In the NCERT book Ganita Manjari: Part II, Chapter 13, Two Variables, One Line.
Revision notes
Linear Equations in Two Variables: revision notes
1. Linear equations in two variables
An equation that can be written \(ax + by + c = 0\), with \(a, b, c\) real and \(a, b\) not both zero, is a linear equation in two variables. Examples: \(2x + 3y = 12\) (\(a = 2, b = 3, c = -12\)); \(x = 4\) is \(1 \cdot x + 0 \cdot y - 4 = 0\).
2. Solutions
A solution is an ordered pair \((x, y)\) that makes the equation true. A linear equation in two variables has infinitely many solutions: choose any \(x\), then solve for \(y\). To test a point, substitute both coordinates.
3. The graph is a straight line
Every solution is a point on one straight line, and every point on that line is a solution. To draw it, make a table of two or three solutions (the intercepts are easiest) and join them.
- \(x\)-intercept: put \(y = 0\); \(y\)-intercept: put \(x = 0\).
- \(y = mx\) passes through the origin.
4. Lines parallel to the axes
\(x = a\) is a line parallel to the \(y\)-axis (vertical), at distance \(|a|\) from it; \(y = b\) is parallel to the \(x\)-axis (horizontal). \(x = 0\) is the \(y\)-axis and \(y = 0\) is the \(x\)-axis. In one variable, \(x = a\) is a single point on the number line; in the plane it is a whole line.
5. Forming equations from situations
Name the two unknowns with units, then write one relation: "\(2\) pens and \(3\) notebooks cost ₹120" gives \(2x + 3y = 120\). A formula such as \(F = \dfrac95C + 32\) is also a linear equation in two variables.
Worked example 1
Is \((4, -1)\) a solution of \(x + 2y = 2\)?
\(4 + 2(-1) = 2\): yes.
Worked example 2
Find the points where \(3x - 2y = 6\) meets the axes.
\(y = 0\): \(x = 2\), point \((2, 0)\); \(x = 0\): \(y = -3\), point \((0, -3)\).
Worked example 3
Find \(k\) if \((1, 3)\) lies on \(kx - y = 5\).
\(k - 3 = 5 \Rightarrow k = 8\).
Common errors
- Substituting \(x\) and \(y\) the wrong way round when testing a point.
- Saying the equation has one solution (or two): it has infinitely many.
- Drawing \(x = 3\) as a horizontal line: it is vertical.
- Forgetting the units or what \(x\) and \(y\) mean in a word problem.
Exam tips
- For a graph, give a table of at least three points: a third point checks the first two.
- Always state what \(x\) and \(y\) stand for before writing the equation.
Topics in this chapter: Solutions · Lines parallel to the axes · The graph is a straight line · Linear equations in two variables · Forming equations from situations.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You can write an equation in the form ax + by + c = 0, test a point and describe lines parallel to the axes.
Step 2
Exam standard
You can find solutions and intercepts, find an unknown coefficient and form an equation from a situation.
Step 3
Full marks on long answers
You can build a linear model from a story, tabulate it and answer each part of a case study.
Step 4
95+ stretch (HOTS)
You can find where lines meet the axes and each other, test collinearity and find areas cut off by lines.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 5 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
The graph of \(x = 4\) in the \(xy\)-plane is
- (a)a line parallel to the \(x\)-axis
- (b)a line through the origin
- (c)a single point
- (d)a line parallel to the \(y\)-axis
If \((2, k)\) lies on the line \(3x - y = 1\), then \(k\) is
- (a)\(-5\)
- (b)\(5\)
- (c)\(7\)
- (d)\(1\)
Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I110 · Problem I117 · Problem I124 · Problem I27. All 29 →
Where marks are lost in Linear Equations in Two Variables
- Point tested with coordinates swapped. Fix: x is the first number, y the second; substitute both.
- 'The equation has a unique solution.' Fix: a single linear equation in two variables has infinitely many solutions.
- x = a drawn horizontally. Fix: x = a is vertical (parallel to the y-axis); y = b is horizontal.
- Intercepts found by setting the wrong variable to zero. Fix: x-intercept: y = 0; y-intercept: x = 0.
- Word-problem equation without defining the variables. Fix: 'Let one pen cost ₹x and one notebook ₹y.'
Test Linear Equations in Two Variables against the clock
Want it against the clock? Take a timed 30-mark chapter test on Linear Equations in Two Variables, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Essentials or the free trial).