Linear Equations in Two Variables

In one variable, "linear" means the variable appears to the first power only , no squares, no roots, no $1/x$ . With two variables it is the same idea, just doubled. A linear equation in two variables can mix $x$ and $y$ , but only as first-degree terms (no $x^2$ , no $xy$ , no $\sqrt{y}$ ). This lesson sets up the precise definition, the standard form, and a few translation exercises.

Definitions

A linear equation in two variables $x$ and $y$ is an equation that can be written in the standard form $ax + by + c = 0,$ where $a, b, c$ are real-number constants and $a, b$ are not both zero. (If both $a$ and $b$ were zero, the equation would degenerate to $c = 0$ , not an equation in $x$ and $y$ at all.)

A solution of the equation is an ordered pair $(p, q)$ for which $ap + bq + c = 0$ . Each solution is one point in the plane.

Concept and form

The standard form is flexible. The form $ax + by + c = 0$ looks rigid, but every linear equation can be rewritten into it. Take $2x + 3y = 6$ . Rearrange: $2x + 3y - 6 = 0$ , so $a = 2, b = 3, c = -6$ . Take $y = 3x - 7$ . Rearrange: $3x - y - 7 = 0$ (multiply through by $-1$ if you prefer leading sign positive: $-3x + y + 7 = 0$ , so $a = -3, b = 1, c = 7$ ). Same equation, different bookkeeping.

Why these constants are not unique. Multiplying the entire equation $ax + by + c = 0$ by a non-zero constant $k$ gives $kax + kby + kc = 0$ , which describes the same equation (same solutions). So the triple $(a, b, c)$ is determined only up to a non-zero scalar multiple. We usually choose the smallest integer triple or the "cleanest" form.

Which equations are linear? The defining property is: every term involves $x, y$ at most to the first power. Forbidden forms:

$x^2 + y = 1$ : has $x^2$ , so not linear.
$xy = 5$ : the term $xy$ is degree $2$ .
$\sqrt{x} + y = 3$ : fractional power.
$\dfrac{1}{x} + y = 0$ : negative power.

Allowed forms (all linear):

$3x + 5y = 7$ .
$x = 4$ (here $a = 1, b = 0, c = -4$ ).
$y = -2$ (here $a = 0, b = 1, c = 2$ ).
$\dfrac{x}{2} + \dfrac{y}{3} = 1$ (clear fractions: $3x + 2y = 6$ ).

One variable in disguise. When $b = 0$ , the equation becomes $ax + c = 0$ , i.e. $x = -\dfrac{c}{a}$ . Even though no $y$ is written, this is still a linear equation in two variables , its solution set is all pairs $(p, q)$ with $p = -\dfrac{c}{a}$ and any $q$ . Geometrically, this is a vertical line. We will return to this in lesson 4.

Translating words into equations. A common exam skill is converting a verbal description into a linear equation.

Sentence: "The cost of a notebook ( $x$ ) is twice the cost of a pen ( $y$ )." Equation: $x = 2y$ , or in standard form $x - 2y = 0$ .

Sentence: "The sum of two numbers is $30$ ." Equation: $x + y = 30$ , or $x + y - 30 = 0$ .

Sentence: "If Anjali had $5$ more rupees and Rajiv had $7$ more rupees, they would each have the same amount, $\text{Rs.}~50$ ." Two separate linear equations: $x + 5 = 50$ (Anjali) and $y + 7 = 50$ (Rajiv). Each is itself a linear equation in two variables, with one variable trivially absent.

The "linear in $x$ alone" trap. Recognise that " $x = 4$ " is a fully valid linear equation in two variables. It just happens that $y$ is free to be anything.

Worked examples

Example 1. Express $3x + 2 = 7y$ in standard form.

Subtract $7y$ : $3x - 7y + 2 = 0$ . So $a = 3, b = -7, c = 2$ .

Example 2. Is $xy = 4$ a linear equation? Why?

No. The term $xy$ has total degree $2$ , not $1$ . The equation is not linear.

Example 3. Write the equation of a line whose every solution has $x = 3$ .

The equation is $x = 3$ , or in standard form $x - 3 = 0$ .

Example 4. Translate: "Twice the cost of an apple ( $x$ ) plus thrice the cost of a banana ( $y$ ) is $\text{Rs.}~30$ ."

$2x + 3y = 30$ , or $2x + 3y - 30 = 0$ .

Example 5. Rewrite $\dfrac{x}{2} - \dfrac{y}{3} = 1$ in standard form with integer coefficients.

Multiply through by $6$ : $3x - 2y = 6$ . Standard form: $3x - 2y - 6 = 0$ .

Try it yourself

Rewrite $5x = 3 + 4y$ in standard form. State $a, b, c$ .
Which are linear in two variables? $3x + 5y = 7$ , $xy = 1$ , $x^2 + y = 2$ , $\sqrt{2}\,x + y = 5$ , $y = 0$ .
Translate: "The age of a father ( $x$ ) is twice the age of his son ( $y$ )."
Translate: "The cost of $3$ pens ( $x$ each) and $4$ pencils ( $y$ each) is $\text{Rs.}~85$ ."
Rewrite $y = 2x - 3$ in standard form.
Rewrite $\dfrac{x}{4} + \dfrac{y}{5} = 1$ in standard form with integer coefficients.
Write a linear equation in two variables whose graph is parallel to the $y$ -axis at $x = -2$ .
Write a linear equation in two variables whose graph passes through $(0, 0)$ .
Is $3x + 5 = 0$ a linear equation in two variables? Justify.
Find $a, b, c$ for the standard form of $2x + y - 4 = 0$ .

Pitfalls / Insight

A constant alone on one side is fine. $3x + 2 = 7y$ is linear; just rearrange to $3x - 7y + 2 = 0$ .
Coefficients can be irrational. $\sqrt{2}\,x + y = 5$ is linear.
" $x = 4$ " is linear in two variables. It just means $b = 0$ . The solution set is a line, namely the vertical line $x = 4$ .

Insight. Once you can spot a linear equation, recognise its standard form, and rearrange one into the other, you have unlocked the gate. The next lesson finds the solutions; the one after that draws the graph. All three are the same equation, viewed differently.