Linear equations in two variables

A puzzle with two unknowns

You buy 2 pens and 1 notebook. You pay 10 dollars. How much does one pen cost, and how much does the notebook cost?

We do not know either price. So we use letters. Let one pen cost $x$ dollars, and let the notebook cost $y$ dollars. The money that you paid gives an equation.

This is an equation with two variables, $x$ and $y$. It is called linear because $x$ and $y$ both have the power 1. There is no $x^2$, no $xy$ and no $\sqrt{x}$.

The standard form

Every linear equation in two variables can be written in the form

Here $a$, $b$ and $c$ are numbers, and $a$ and $b$ are not both zero. Our equation $2x + y = 10$ becomes $2x + y - 10 = 0$. So $a = 2$, $b = 1$ and $c = -10$.

Some equations may look different, but are linear. $y = 3x + 2$ becomes $3x - y + 2 = 0$. The equation $x = 4$ is also linear in two variables, because we can write it as $x + 0 \cdot y - 4 = 0$, with $b = 0$.

An equation such as $x^2 + y = 3$ or $xy = 6$ is not linear.

What is a solution?

A solution is a pair of values, one for $x$ and one for $y$, that makes the equation true. We write it as $(x, y)$.

Is $(3, 4)$ a solution of $2x + y = 10$? We put $x = 3$ and $y = 4$ in the left side.

It is 10, the same as the right side. So yes, $(3, 4)$ is a solution. A pen costs 3 dollars, and the notebook costs 4 dollars.

Is $(1, 7)$ a solution? $2 \times 1 + 7 = 9$. This is not 10. So no.

There are many solutions

Try $(2, 6)$. $2 \times 2 + 6 = 10$. It works. Try $(4, 2)$. $8 + 2 = 10$. It works too.

An equation with one unknown, like $2x = 10$, has one solution. But an equation with two unknowns has infinitely many solutions. We can choose any value for $x$. Then $y$ is decided.

You should find that the dot never leaves the line, and the left side always equals the right side. Every place the dot stops is a solution.

How to find solutions

We choose a value of $x$. Then we want $y$, so we write the equation as $y = 10 - 2x$.

In the shop, only some of these solutions make sense. A price cannot be negative. So $x = 6$, which gives $y = -2$, is a solution of the equation, but not of the shop problem.

More examples from real life

The perimeter of a rectangle. A rectangle has a perimeter of $20\ \text{cm}$. Let the sides be $x$ and $y$. Then $2x + 2y = 20$, or $x + y = 10$. There are many such rectangles: $(1, 9)$, $(4, 6)$, $(5, 5)$, and so on. Here $x$ and $y$ must be positive, so the solutions with $x \le 0$ do not fit the problem.

Temperature. The relation between degrees celsius $C$ and degrees fahrenheit $F$ is $F = \dfrac{9}{5}C + 32$. This is a linear equation in two variables. A solution is $(C, F) = (0, 32)$, which is the freezing point of water. Another is $(100, 212)$, the boiling point.

Distance. A taxi charges 30 dollars to start and 12 dollars for each kilometre. The cost $y$ for $x$ kilometres is $y = 30 + 12x$. When $x = 5$, $y = 90$. A solution is $(5, 90)$.

A way to check

To find whether a pair is a solution, put it in the equation and compare the two sides. Do not guess. If the two sides are equal, it is a solution. If not, it is not.

Practise with new numbers

Try it in Python

This program takes $x$ from 0 to 5 and finds $y$ for the equation $2x + y = 10$. It also checks every pair.

Common mistakes

• Thinking that there is only one solution. A linear equation in two variables has infinitely many.
• Checking only one side of the equation. Compute the left side, and compare with the right.
• Putting $x$ and $y$ in the wrong order. The pair $(3, 4)$ means $x = 3$, $y = 4$. The pair $(4, 3)$ is a different pair.
• Ignoring the problem. A solution of the equation may not suit the story, such as a negative price.

Remember these

• A linear equation in two variables has the form $ax + by + c = 0$.
• A solution is a pair $(x, y)$ that makes the equation true.
• There are infinitely many solutions. Choose $x$, and then find $y$.
• To check a pair, put it in the equation.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $(2, 3)$ a solution of $x + 2y = 8$?

2. Write $3x = 5y + 1$ in the form $ax + by + c = 0$. What are $a$, $b$ and $c$?

3. Find three solutions of $x + y = 7$.

4. A pen costs $x$ dollars and a book costs $y$ dollars. Write the equation for: $3$ pens and $2$ books cost $17$ dollars.