Chapter 13 Class 9 - Two Variables, One Line (Ganita Manjari)

Study Chapter 13: Two Variables, One Line from NCERT Ganita Manjari, Class 9 Maths, Part II, with Teachoo. Learn about linear equations in two variables, ordered-pair solutions, straight-line graphs, slope and intercepts, and methods for solving a pair of linear equations.

How can one equation have infinitely many solutions? What does the slope of a line tell you? When do two equations have one common solution, no solution or infinitely many solutions?

Teachoo helps you connect equations with their graphs, so that each calculation has a clear meaning. Begin with identifying solutions, move to plotting lines, and then learn how substitution, elimination and graphical methods find the values that satisfy two equations together.

Whether you are studying the chapter for the first time, practising Exercise Sets 13.1–13.5, or revising the End-of-Chapter Exercises, make Teachoo your study companion for learning linear equations with confidence.

What will you learn in Chapter 13?

Linear equations in two variables

Learn to express relationships using two variables and write equations in the standard form:

ax + by + c = 0

Here, a, b and c are real numbers, and a and b are not both zero. Identify the coefficients and constant term, including equations containing fractions or decimals.

Solutions as ordered pairs

Understand why a solution is an ordered pair (x, y). Check a proposed solution by substitution and find additional pairs that satisfy the equation.

Graphs of linear equations

Plot solutions on the coordinate plane and understand why the graph is a straight line. Connect each point on the line with a solution of the equation.

Slope and intercepts

Learn what slope tells you about the change in y relative to the change in x. Study the slope-intercept form:

y = mx + d

Here, m is the slope and d is the y-intercept.

Pairs of linear equations

Understand what it means for an ordered pair to satisfy two equations simultaneously. Explore intersecting, parallel and coincident lines.

Substitution, elimination and graphical methods

Learn to solve a pair of equations algebraically or locate its common solution graphically.

Word problems

Translate situations involving prices, ages, digits, fares, fractions and other quantities into equations. Interpret your answer in the original context.

Teachoo helps you move between words, equations, tables and graphs, building a connected understanding of the chapter.

Understand a linear equation with a simple example

Consider:

2x + y = 7

Some solutions are:

• (0, 7) because 2 × 0 + 7 = 7.

• (1, 5) because 2 × 1 + 5 = 7.

• (2, 3) because 2 × 2 + 3 = 7.

These ordered pairs represent points on the same straight line.

Rewriting the equation gives:

y = −2x + 7

Therefore, its slope is −2 and its y-intercept is 7. The line crosses the y-axis at (0, 7).

With Teachoo, learn to connect the equation, its solutions and its graph instead of treating them as separate topics.

How do you solve a pair of linear equations?

A common solution must satisfy both equations.

For example:

x + y = 7
x − y = 1

Adding the equations eliminates y:

2x = 8
x = 4

Substitute x = 4 into x + y = 7:

4 + y = 7
y = 3

Therefore, the common solution is (4, 3). Graphically, this is the point where the two lines intersect.

Teachoo helps you understand why a method works, choose a convenient approach and check the answer in both original equations.

Exercise Sets 13.1–13.5 and End-of-Chapter Exercises

Ganita Manjari Class 9 Chapter 13 contains five exercise sets—13.1, 13.2, 13.3, 13.4 and 13.5—followed by 16 numbered End-of-Chapter Exercises. Some questions contain several subparts, and the final five end-of-chapter questions are starred.

The chapter includes questions on:

• Writing equations in standard form.

• Identifying coefficients and constant terms.

• Checking and finding ordered-pair solutions.

• Drawing and interpreting straight-line graphs.

• Calculating slopes and identifying intercepts.

• Solving pairs of equations.

• Determining whether lines intersect, are parallel or coincide.

• Modelling everyday situations using two unknowns.

• Interpreting rates, temperature conversions and plan comparisons.

• Finding parameter values for different types of solutions.

Use Teachoo to understand the method, then attempt questions independently and verify your results.

Why study Two Variables, One Line with Teachoo?

This chapter brings algebra and coordinate geometry together. A sign error can change an equation, reversing an ordered pair can change a point, and a graph can reveal whether two equations have a common solution.

Teachoo teaches these connections clearly, helping you understand what each step means and how to check it. Define your variables, form the equations, solve carefully and interpret the result.

You do not need to master every method at once. Start with one equation and its solutions, then build towards solving a pair. Choose Teachoo as your go-to learning platform for Ganita Manjari Class 9 Maths, and develop confidence through clear explanations and purposeful practice.

Frequently asked questions

1. What is Chapter 13 in Class 9 Ganita Manjari?

Chapter 13 in Ganita Manjari, Class 9 Maths, Part II, is “Two Variables, One Line.” It covers linear equations in two variables, solutions and graphs, slope-intercept form, and pairs of linear equations. Teachoo helps you understand these ideas and apply them to textbook questions.

2. What is a linear equation in two variables?

A linear equation in two variables can be written as ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero.

For example, 3x + 2y − 12 = 0 is a linear equation in two variables. Its graph is a straight line.

3. What is a solution of a linear equation in two variables?

A solution is an ordered pair (x, y) that makes the equation true when substituted.

For example, (2, 3) satisfies 3x + 2y = 12, because 3 × 2 + 2 × 3 = 12.

4. Why does the order in an ordered pair matter?

In (x, y), the first value belongs to x and the second belongs to y. Reversing them generally changes both the point and whether it satisfies an equation.

For example, (2, 3) satisfies 3x + 2y = 12, but (3, 2) does not.

5. How many solutions does one linear equation in two variables have?

Over the real numbers, a linear equation in two variables has infinitely many solutions. These solutions form a straight line.

A word problem may impose additional restrictions, such as requiring whole numbers or non-negative quantities, which can limit the acceptable solutions.

6. How do you draw the graph of a linear equation?

Find two distinct ordered-pair solutions, plot them accurately and draw the straight line through them. You can check your graph by plotting a third solution.

Teachoo helps you choose convenient values and connect the plotted points with the original equation.

7. What is the slope of a line?

For two distinct points on a non-vertical line:

Slope = change in y ÷ change in x
m = (y₂ − y₁) ÷ (x₂ − x₁)

A positive slope means the line rises from left to right; a negative slope means it falls. A horizontal line has slope zero, while a vertical line has undefined slope.

8. What is slope-intercept form?

Slope-intercept form is y = mx + d, where m is the slope and d is the y-intercept.

For example, in y = 3x + 2, the slope is 3, and the line crosses the y-axis at (0, 2). Vertical lines cannot be written in this form.

9. What are x-intercepts and y-intercepts?

An x-intercept is where a line meets the x-axis, so y = 0.

A y-intercept is where it meets the y-axis, so x = 0.

Substituting these values into the equation helps you find the intercepts, when they exist.

10. What is a pair of linear equations in two variables?

It is a set of two linear equations involving the same variables. A common solution must satisfy both equations.

Graphically, a common solution corresponds to a point lying on both lines.

11. When does a pair of linear equations have one solution, no solution or infinitely many solutions?

• Intersecting lines: One common point, so one unique solution.

• Distinct parallel lines: No common point, so no solution.

• Coincident lines: The same line, so infinitely many solutions.

Teachoo helps you connect these graphical cases with the algebraic relationships between the equations.

12. What is the substitution method?

In substitution, express one variable in terms of the other using one equation. Substitute that expression into the second equation to obtain an equation in one variable.

Solve it, then substitute back to find the remaining variable and check both original equations.

13. What is the elimination method?

In elimination, add or subtract suitable multiples of the equations so that one variable cancels.

Solve the resulting equation in one variable, then substitute back to find the other. This method is especially convenient when coefficients already match or can easily be made equal.

14. How do you form equations from a word problem?

Define the two unknown quantities clearly, including their units. Translate each independent relationship into an equation.

After solving, check that the values satisfy both relationships and make sense in the situation. Teachoo helps you practise this process with problems involving costs, ages, digits, fares and fractions.

15. Which exercises are included in Chapter 13?

The chapter includes Exercise Sets 13.1, 13.2, 13.3, 13.4 and 13.5, followed by 16 numbered End-of-Chapter Exercises. The exercises cover equations, graphs, slopes, simultaneous solutions and applications.

16. Why choose Teachoo for Two Variables, One Line?

Choose Teachoo to understand how an equation becomes a graph, how slope describes a relationship, and how two equations can determine a common solution. Teachoo’s focus on clear reasoning makes it a strong choice for studying Ganita Manjari Class 9 Chapter 13, practising textbook questions and revising confidently.

Learn Two Variables, One Line with Teachoo—understand the equation, read the graph and solve with confidence.