What is a pair of linear equations?
A linear equation in two variables looks like ax + by = c. Here x and y are unknown numbers (variables). a, b and c are fixed numbers, and a and b are not both zero.
"Linear" means its graph is a straight line. One such equation has endless answers. Two equations about the same x and y are called a pair. We want the one (x, y) that makes both true together.
Graph of one linear equation
Take one rule, like 2x + y = 6. It has many answers. Each answer is a pair (x, y).
- Pick an easy x, like 0. Then y = 6. Point (0, 6).
- Pick x = 3. Then y = 0. Point (3, 0).
- Pick one more to check, x = 1. Then y = 4. Point (1, 4).
- Plot the points and join them with a ruler.
All the points sit on one straight line. Every point on the line is an answer. A point off the line is not an answer. Look at step 1 of the 3D: the dots for x + y = 5 all line up.
Special lines: x = 2 is a straight up-and-down line. y = 3 is a flat line. y = 2x passes through (0, 0).
Slope-intercept form: y = mx + c
We can rewrite any line (with b ≠ 0) so that y is alone: y = mx + c.
- m is the slope. Slope means steepness. When x goes up by 1, y goes up by m.
- c is the y-intercept. It is where the line cuts the y-axis, at the point (0, c).
Example: 2x + y = 6 becomes y = −2x + 6. Slope −2: the line goes down 2 for every 1 step right. It cuts the y-axis at 6.
Why this helps: two lines with the same slope but different c are parallel (no answer). Same slope and same c means the same line. Different slopes means the lines cross once. In the free-play step, change the green line and watch its slope and intercept.
Graphical method
Step 1. For each equation, make a small table: pick 2 or 3 values of x and work out y.
Step 2. Plot the points and join them with a ruler. You get two lines.
Step 3. Read where the lines meet. That point (x, y) is the solution.
Example: x + y = 5 gives (0,5), (5,0). x − y = 1 gives (1,0), (3,2). The lines meet at (3, 2), so x = 3, y = 2. Try it in the 3D above (steps 1 to 3).
Three possible pictures: consistency
- Lines cross → exactly one solution. The pair is consistent.
- Lines are parallel → no solution. The pair is inconsistent.
- Lines coincide (lie on top of each other) → infinitely many solutions. The pair is consistent and dependent.
Ratio test without drawing
For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compare the ratios:
- a₁/a₂ ≠ b₁/b₂ → lines cross → one solution.
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel → no solution.
- a₁/a₂ = b₁/b₂ = c₁/c₂ → same line → infinitely many.
Why? If a and b are in the same ratio, the lines lean at the same angle (same slope). Then c decides whether they are the same line or just side by side.
Substitution method
"Substitute" means "put in place of".
- From one equation, write one variable in terms of the other. Example: x − y = 1 gives x = y + 1.
- Put this into the other equation: (y + 1) + y = 5, so 2y = 4, y = 2.
- Put y back: x = 2 + 1 = 3.
- Check in both equations: 3 + 2 = 5 ✓, 3 − 2 = 1 ✓.
If the variable vanishes and you get something always true (like 0 = 0), there are endless solutions. If you get something false (like 0 = 3), there is no solution.
Elimination method
"Eliminate" means "remove".
- Multiply one or both equations so that one variable has the same number in front (same coefficient).
- Add or subtract the equations. That variable disappears.
- Solve the simple equation left over.
- Put the value back to find the other variable, then check.
Example: x + y = 5 and x − y = 1. Adding them: 2x = 6, so x = 3; then y = 2. In the 3D free-play step, the button shows this line by line for any pair you make.
Word problems
Four steps turn a story into maths:
- Name the two unknowns: "Let the price of a pen be ₹x and a notebook be ₹y."
- Write one equation for each fact in the story.
- Solve by substitution or elimination.
- Check the answer makes sense (prices positive, ages whole numbers) and answer in words.
Common types: prices, ages, two-digit numbers, fractions, speed of boat and stream, fixed charge plus rate, and angles or sides of a figure.
Try it: a kitchen experiment
Ask someone at home to secretly pick two numbers. Ask only for their sum and their difference. Write x + y = sum and x − y = difference. Add the two equations: 2x = sum + difference. Find x, then y. Surprise them! Then in the 3D free play, set the green line to x − y = your difference and see the meeting point move.
Key formulas and definitions
- One equation ax + by = c: its graph is a straight line; every point on it is a solution
- Slope-intercept form: y = mx + c (m = slope, c = y-intercept)
- Slope from ax + by = c: m = −a/b, intercept c/b
- General form: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
- a₁/a₂ ≠ b₁/b₂ → one solution (intersecting, consistent)
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution (parallel, inconsistent)
- a₁/a₂ = b₁/b₂ = c₁/c₂ → infinitely many (coincident, dependent)
- Substitution: express one variable, put it in the other equation
- Elimination: make coefficients equal, add or subtract
Worked examples
1. Draw the graph of 2x + y = 6. Is (2, 2) a solution?
Table: x = 0 → y = 6; x = 3 → y = 0; x = 1 → y = 4. Plot (0,6), (3,0), (1,4) and join. Check (2, 2): 2×2 + 2 = 6 ✓, so yes, it lies on the line.
2. Write 3x + 2y = 8 in the form y = mx + c. Find the slope and y-intercept.
2y = −3x + 8, so y = −(3/2)x + 4. Slope m = −3/2 (goes down 3 for every 2 steps right). y-intercept c = 4, point (0, 4).
3. Solve by elimination: x + y = 7, x − y = 3.
Add: 2x = 10, so x = 5. Then y = 7 − 5 = 2. Check: 5 − 2 = 3 ✓. Answer x = 5, y = 2.
4. Solve by substitution: y = 2x and 3x + y = 15.
Put y = 2x: 3x + 2x = 15, so 5x = 15, x = 3. Then y = 6. Check: 9 + 6 = 15 ✓.
5. Without solving, say how many solutions 2x + 3y = 6 and 4x + 6y = 15 have.
a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 6/15 = 2/5. The first two are equal but the third differs, so the lines are parallel: no solution (inconsistent).
6. Find k so that kx + 2y = 5 and 3x + y = 1 have exactly one solution.
One solution needs a₁/a₂ ≠ b₁/b₂: k/3 ≠ 2/1, so k ≠ 6. Any k except 6 works.
7. Solve by elimination: 3x + 2y = 11 and 2x + 3y = 4.
Multiply the first by 3 and the second by 2: 9x + 6y = 33 and 4x + 6y = 8. Subtract: 5x = 25, x = 5. Put back: 15 + 2y = 11, so 2y = −4, y = −2. Check: 10 − 6 = 4 ✓.
8. 2 pens and 3 notebooks cost ₹80. 1 pen and 1 notebook cost ₹30. Find each price.
Let pen = ₹x, notebook = ₹y. 2x + 3y = 80 and x + y = 30. From the second, x = 30 − y. Then 2(30 − y) + 3y = 80 → 60 + y = 80 → y = 20, x = 10. A pen costs ₹10, a notebook ₹20.
9. A taxi charges a fixed amount plus a rate per km. 10 km costs ₹105 and 15 km costs ₹155. Find the fixed charge and the rate.
Let fixed = ₹x, rate = ₹y per km. x + 10y = 105 and x + 15y = 155. Subtract: 5y = 50, y = 10. Then x = 105 − 100 = 5. Fixed charge ₹5, rate ₹10 per km.
10. A boat goes 30 km downstream in 2 hours and comes back upstream in 3 hours. Find the speed of the boat in still water and of the stream.
Let boat = x km/h, stream = y km/h. Downstream speed x + y = 30/2 = 15; upstream x − y = 30/3 = 10. Add: 2x = 25, x = 12.5 km/h; y = 2.5 km/h.
Common mistakes
- Forgetting to multiply every term (including the number on the right side) when making coefficients equal.
- Adding when you should subtract: subtract if the coefficients have the same sign, add if they have opposite signs.
- Mixing up the cases: equal a and b ratios mean parallel or same line, never one solution.
- Stopping after finding one variable, or not checking the answer in both equations.