Two Variables, One Line
One equation, infinitely many pairs — and every pair plots onto a single straight line. Two lines then tell you everything about their solutions.
Original notes — ready CBSE (NCERT, English Medium) • Class 9 • Mathematics • Book: Ganita Manjari — Part 2
Chapter 13: Two Variables, One Line — EG's own notes, in simple words
Book: Ganita Manjari — Part 2 (CBSE, Class 9) | Chapter 13 | Medium: English
Diagram (labels are printed on the image): A straight line graph crossing both axes, with three solution points marked on the line and one point off the line that is not a solution.
Drawing a straight road with a ruler — every point on it follows the same rule.
What the chapter is about
One variable gives points on a line; two variables give a whole relationship. The equation 3x + y = 9 has not one answer but infinitely many pairs (x, y) that satisfy it: (3, 0), (2, 3), (0, 9) and many more. The beautiful discovery of this chapter is that when you plot all those pairs, they lie on one single straight line.
That is why the equation is called a linear equation in two variables. Its graph is a line, and every point on the line is a solution, while every point off the line is not. Finding solutions is easy — choose a value for x, and the equation tells you y — and two solutions are enough to draw the line.
Pairs of such equations appear next: two lines may cross at one point (one common solution), run parallel (no solution), or coincide (infinitely many solutions). Reading that picture is the first step towards solving equations graphically, a skill used in science and economics alike.
The chapter writes lines in slope-intercept form, y = mx + d. Here m is the slope — how much y rises when x rises by 1 — and d is the intercept, the length cut off on the y-axis from the origin, the value of y when x = 0. So 2x + y = 6 becomes y = −2x + 6 at once: slope −2, intercept 6, and two easy points (0, 6) and (3, 0) draw the line. Reading a line this way is faster than finding points one by one.
Slope also settles pairs of lines without drawing them. Lines with the same slope but different intercepts are parallel and never meet, so their equations have no common solution. Lines with different slopes cross exactly once, giving one common solution. And if one equation is just a multiple of the other, the lines coincide and every solution is common. The chapter's graphical method — drawing both lines and reading their meeting point — is the picture behind all three cases.
What you should learn
- Show that solutions of a linear equation in two variables form a straight line.
- Write lines in slope-intercept form y = mx + d and read slope and intercept.
- Decide from slopes whether two lines meet once, never, or coincide.
- Solve a pair of equations graphically by reading their meeting point.
Key Terms
- Linear equation in two variables — An equation of the form ax + by + c = 0, with x and y to the first power only.
- Solution — A pair (x, y) that makes the equation true.
- Graph — The picture formed by plotting all solutions of an equation; for a linear equation it is a straight line.
- Slope — A measure of a line's steepness — how much y changes when x increases by 1.
- System of equations — Two or more equations considered together, sharing the same variables.
- Slope-intercept form — The form y = mx + d for a line, where m is the slope and d is the intercept on the y-axis.
- Parallel lines — Lines with equal slopes and different intercepts; they never meet, so their equations have no common solution.
Remember
- A linear equation in two variables has infinitely many solutions.
- Two solutions are enough to draw its graph, a straight line.
- Every point on the line is a solution; points off the line are not.
- Two lines: crossing = one solution, parallel = none, same line = infinitely many.
- In y = mx + d, m is the slope and d is the y-intercept — read both before plotting.
- Same slope, different intercept: parallel. Same slope, same intercept: the same line.
Exam Tip / Common Mistake
Common mistake: stopping after finding one solution and thinking the equation is 'solved'. A two-variable equation needs pairs — find at least two, and check each in the equation.
Chart / Table: Solutions of 3x + y = 9
| Point | Detail |
|---|---|
| x = 0 | y = 9 → point (0, 9) |
| x = 1 | y = 6 → point (1, 6) |
| x = 2 | y = 3 → point (2, 3) |
| x = 3 | y = 0 → point (3, 0) |
| All four points | Lie on the same straight line |
Worked method: rewriting 2x + y = 6 in slope-intercept form
- Start with 2x + y = 6. Move 2x to the other side: y = −2x + 6.
- Read off: slope m = −2, intercept d = 6. The line falls 2 for every 1 step right.
- Two points draw it: at x = 0, y = 6, giving (0, 6); at x = 3, y = 0, giving (3, 0).
Practice set — try it yourself
- Find three solutions of x + y = 7.
Hint: Hint: Choose x values and find y from the equation. - Is (2, 5) a solution of 3x + y = 9? Check.
Hint: Hint: Put x = 2 and y = 5 into the left side. - Find where the line 3x + y = 9 meets the x-axis.
Hint: Hint: On the x-axis, y = 0. - Draw the graph of x − y = 2 using two solutions.
Hint: Hint: Pick easy x values, such as 0 and 2. - Two lines are parallel. How many common solutions do their equations have?
Hint: Hint: Parallel lines never meet.
Questions with Answers
- What is a linear equation in two variables?
An equation of the form ax + by + c = 0 where a and b are not both zero. It involves two variables, each to the first power. - Find two solutions of 2x + y = 8.
Take x = 0 → y = 8, giving (0, 8). Take x = 4 → y = 0, giving (4, 0). Both satisfy the equation. - Why is the graph of a linear equation a straight line?
Because every solution pair (x, y) plots onto the same straight line — the equation fixes a constant relationship between x and y, which is exactly what a line shows. - What does it mean if two lines coincide?
The two equations have the same graph, so every solution of one is a solution of the other — infinitely many common solutions. - How can you check a solution?
Substitute the pair into the equation. If the left side equals the right side, the pair is a solution; otherwise it is not.
Extension questions
- Write 2x + y = 6 in slope-intercept form. State its slope and intercept.
y = −2x + 6; slope −2, intercept 6. - Are y = 2x + 1 and y = 2x − 5 parallel? Give a reason.
Yes — both have slope 2 but different intercepts, so they never meet. - Solve x + y = 5 and x − y = 1 graphically, in words.
The lines cross at (3, 2): adding the equations gives 2x = 6, so x = 3 and y = 2, and (3, 2) satisfies both. - Do 2x + 2y = 4 and x + y = 2 have one solution, none, or infinitely many? Why?
Infinitely many: the first equation is twice the second, so both are the same line. - Is (1, 4) a solution of y = 2x + 2? Check.
Yes: 2 × 1 + 2 = 4, which matches y = 4. - Find where the line y = 3x − 6 meets the x-axis and the y-axis.
On the x-axis y = 0, so 3x = 6 and x = 2: point (2, 0). On the y-axis x = 0, so y = −6: point (0, −6).
Every solution pair is a point, all the points are one line, and y = mx + d reads the line aloud: slope m, intercept d, and pairs decided by comparing slopes.
How to use this page: Read the summary, say the Key Terms in your own words, then try the practice set without looking. Use the Questions with Answers to check yourself afterwards. Ask your teacher or parent to check your answers.