Linear Equations in Two Variables

An equation of the form ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero, is called a linear equation in two variables. The word "linear" means the highest power of each variable is 1 — there is no x², y², √x, xy or similar term.

For example, 2x + 3y − 5 = 0 is linear, but x² + y = 4 and xy = 6 are not linear because the degree of a variable exceeds 1 or the two variables are multiplied together.

Even an equation that looks like it has only one variable, such as 2x + 3 = 0 or y − 5 = 0, can be written as a linear equation in two variables by inserting the missing variable with coefficient zero: 2x + 0y + 3 = 0 or 0x + y − 5 = 0.

Worked Example: Write the equation 2x = 5 − 3y in the standard form ax + by + c = 0 and state the values of a, b and c.

Rearranging, 2x + 3y − 5 = 0. Comparing with ax + by + c = 0, we get a = 2, b = 3 and c = −5.

A pair of values, one for x and one for y, which satisfies a linear equation is called a solution of that equation. A solution is written as an ordered pair (x, y), with the x-value always written first.

Because a linear equation in two variables has two unknowns but only one equation connecting them, we can choose any value for one variable and then solve for the other. Since x can take infinitely many values, a linear equation in two variables has infinitely many solutions.

Worked Example: Find four solutions of 2x + y = 6.

• Let x = 0: 2(0) + y = 6 ⟹ y = 6, giving (0, 6)
• Let x = 1: 2(1) + y = 6 ⟹ y = 4, giving (1, 4)
• Let x = 2: 2(2) + y = 6 ⟹ y = 2, giving (2, 2)
• Let x = 3: 2(3) + y = 6 ⟹ y = 0, giving (3, 0)

Worked Example: Is (2, 3) a solution of x − 2y = 4?

Substitute x = 2, y = 3 into the left side: 2 − 2(3) = 2 − 6 = −4, which is not equal to the right side, 4. So (2, 3) is not a solution.

Since a linear equation in two variables has infinitely many solutions, each solution (x, y) can be plotted as a point on the Cartesian plane. When all these points are plotted, they always lie on a single straight line. This line is called the graph of the linear equation.

Two important facts follow from this:

• Every point on the line satisfies the equation, i.e. it is a solution.
• Every solution of the equation lies on the line.
• A point that does not lie on the line is not a solution.

Since two points are enough to fix a straight line, it is good practice to find at least three solutions — the third acts as a check that no arithmetic mistake was made.

Worked Example: Find three solutions of x + 2y = 8 and use them to describe its graph.

• x = 0 ⟹ 2y = 8 ⟹ y = 4, point (0, 4)
• x = 2 ⟹ 2 + 2y = 8 ⟹ y = 3, point (2, 3)
• x = 4 ⟹ 4 + 2y = 8 ⟹ y = 2, point (4, 2)

Plotting (0, 4), (2, 3) and (4, 2) on graph paper and joining them with a ruler gives a straight line extending in both directions — this line is the graph of x + 2y = 8.

Some linear equations involve only one variable but are still linear equations in two variables, because the missing variable is present with coefficient 0.

The equation y = 0 (i.e., 0x + y = 0) is satisfied by every point whose y-coordinate is 0, such as (1, 0), (−3, 0), (5, 0). All such points lie on the x-axis, so y = 0 is the equation of the x-axis.

Similarly, x = 0 is the equation of the y-axis, since it is satisfied by every point of the form (0, y).

More generally, the equation y = k (k a fixed real number) is satisfied by all points (x, k) for any x, so its graph is a line parallel to the x-axis, at a distance |k| from it — above the x-axis if k is positive, below if k is negative.

Likewise, the equation x = k gives a line parallel to the y-axis, at a distance |k| from it — to the right of the y-axis if k is positive, to the left if k is negative.

Worked Example: Describe the graphs of y = 3 and x = −2.

y = 3 is a line parallel to the x-axis, 3 units above it, passing through points such as (0, 3), (1, 3), (−2, 3). x = −2 is a line parallel to the y-axis, 2 units to the left of it, passing through points such as (−2, 0), (−2, 1), (−2, 5).

Many everyday statements that relate two quantities can be written as a linear equation in two variables. The key step is to choose clear variable names for the two quantities before translating the words into an equation.

Worked Example: The cost of a fountain pen is Rs 10 more than twice the cost of a ball pen. Write this as a linear equation in two variables.

Let the cost of the fountain pen be Rs x and the cost of the ball pen be Rs y. "Twice the cost of a ball pen" is 2y, and the fountain pen costs Rs 10 more than this, so x = 2y + 10, which in standard form is x − 2y − 10 = 0.

Worked Example: The relation between temperature in Fahrenheit (F) and Celsius (C) is the linear equation F = (9/5)C + 32. Find F when C = 0, and find C when F = 86.

When C = 0: F = (9/5)(0) + 32 = 32, so 0°C corresponds to 32°F.

When F = 86: 86 = (9/5)C + 32 ⟹ 54 = (9/5)C ⟹ C = 54 × 5/9 = 30. So 86°F corresponds to 30°C.