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Linear equations in two variables

See why ax + by = c draws a straight line, and solve a pair of such equations by substitution, by elimination and by graphing.

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The big idea

An equation like x + y = 4 has two unknowns, and on its own it has endless solutions. But two such equations together usually pin down exactly one pair of values. The picture behind this is two straight lines crossing.

A linear equation in two variables has the form ax + by + c = 0, with a and b not both zero. Each solution is a pair (x, y). For x + y = 4 the pairs (0, 4), (1, 3), (2, 2) and (4, 0) all work, and so do infinitely many others.

If you plot every solution on a graph you get a straight line. That is why the equation is called linear. Two points are enough to draw the line, but plotting a third is a good check.

A pair of linear equations can be solved by elimination: add or subtract the equations so that one variable cancels. Or by substitution: write one variable in terms of the other and substitute it into the second equation.

Graphically, the two lines either cross at one point (one solution, a consistent pair), are parallel (no solution, an inconsistent pair) or are the same line (infinitely many solutions). Compare the ratios a1/a2, b1/b2 and c1/c2 to decide which case you have without drawing.

Two lines crossing at one pointA graph with x and y axes. The line x plus y equals 4 slopes down to the right and the line x minus y equals 2 slopes up to the right. They cross at the point (3, 1), which is marked with a dot.Solving x + y = 4 and x − y = 2 by graph x y 1 2 3 4 5 6 1 2 3 4 5 x + y = 4 x − y = 2 (3, 1) Both equations are true only at (3, 1): 3 + 1 = 4 and 3 − 1 = 2.
Each equation is a straight line. The pair of equations is solved where the two lines meet: (3, 1).

Worked examples

See how it works

  1. Elimination: x + y = 4 and x − y = 2

    Add the two equations: 2x = 6, so x = 3. Put x = 3 into the first: 3 + y = 4, so y = 1. The solution is (3, 1). Check in the second: 3 − 1 = 2. Correct.

  2. Substitution: 2x + 3y = 12 and x = y + 1

    Replace x by y + 1 in the first equation: 2(y + 1) + 3y = 12, so 2y + 2 + 3y = 12, 5y = 10 and y = 2. Then x = 2 + 1 = 3. Check: 2 × 3 + 3 × 2 = 12.

  3. A word problem

    Five pencils and seven pens cost 50 rupees; seven pencils and five pens cost 46 rupees. Let a pencil cost x and a pen cost y: 5x + 7y = 50 and 7x + 5y = 46. Adding gives 12x + 12y = 96, so x + y = 8. Subtracting the first from the second gives 2x − 2y = −4, so x − y = −2. Adding these two: 2x = 6, x = 3 and y = 5. A pencil costs 3 rupees and a pen 5 rupees.

  4. How many solutions?

    x + 2y = 4 and 2x + 4y = 12: the ratios are a1/a2 = 1/2, b1/b2 = 1/2 but c1/c2 = 4/12 = 1/3. The first two match and the third does not, so the lines are parallel and there is no solution.

Try it at home

Plot two lines on a paper grid

You will need

  • A sheet of ruled or squared paper
  • A pencil and ruler
  • Two coloured pencils if you have them

What to do

  1. Draw an x-axis and a y-axis and number them from 0 to 6 using one square per unit.
  2. For x + y = 4, make a table: x = 0 gives y = 4, x = 2 gives y = 2, x = 4 gives y = 0. Plot the three points and join them with a ruler.
  3. For x − y = 2, make a table: x = 2 gives y = 0, x = 4 gives y = 2, x = 6 gives y = 4. Plot and join them in the other colour.
  4. Find the point where the two lines cross and read its coordinates. It should be (3, 1).
  5. Check: 3 + 1 = 4 and 3 − 1 = 2. Both equations are true at the crossing point.

Practice sheet

Check what you learned

Write your answers in a notebook, or print this sheet and write on it. When you finish, mark your work with the answer key.

Practice sheet: Linear equations in two variables (Classes 9-10, Maths)

  1. Is (2, 1) a solution of 3x − y = 5?
  2. Write three solutions of x + y = 5.
  3. Solve x + y = 10 and x − y = 4.
  4. What shape is the graph of a linear equation in two variables?
  5. Solve by substitution: y = 2x and x + y = 9.
  6. Solve 2x + y = 7 and x − y = 2.
  7. How many solutions do x + 2y = 3 and 2x + 4y = 6 have?
  8. How many solutions do x + y = 3 and x + y = 7 have?
  9. The sum of two numbers is 20 and their difference is 4. Find them.
  10. Where does the line x − y = 2 cross the y-axis?

Answer key

Answers: Linear equations in two variables

Do the practice sheet first, then check here. If an answer is different from yours, read the lesson again and try once more.

  1. Yes: 3 × 2 − 1 = 5.
  2. For example (0, 5), (1, 4) and (5, 0). Any pair that adds to 5 is correct.
  3. Adding gives 2x = 14, so x = 7 and y = 3.
  4. A straight line.
  5. x + 2x = 9, so x = 3 and y = 6.
  6. Adding gives 3x = 9, so x = 3; then y = 1.
  7. Infinitely many: the second equation is twice the first, so both describe the same line.
  8. None: the lines are parallel, because x + y cannot be both 3 and 7.
  9. 12 and 8: x + y = 20 and x − y = 4 give 2x = 24.
  10. On the y-axis x = 0, so −y = 2 and y = −2. The point is (0, −2).

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