medium difficulty9 min

Pair of linear equations in two variables

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

In one line: Think of the two equations as two rules and the solution as the one pair that obeys both rules together.

Practice This Concept

Understand it

This chapter studies two equations together because many real problems need two conditions to be satisfied at the same time. The solution is the ordered pair of values of x and y that makes both equations true. In exams, students must learn to identify the type of pair and choose a suitable method quickly.

Example

For example, 2x + y = 7 and x - y = 1 form a pair of linear equations in two variables. Their common solution is the pair that satisfies both equations.

Formula and rule

A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For 2x + y = 7 and x - y = 1, add the equations after rewriting if needed: 2x + y = 7 and x - y = 1.

1Adding gives 3x = 8, so x = 8/3.
2Substituting in x - y = 1 gives y = 5/3.

Method you can reuse

1Write both equations in standard form if needed.
2Decide whether substitution, elimination, or graphing is suitable.
3Solve for one variable.
4Substitute back to get the other variable.
5Verify the answer in both equations.

Visual tip

A graph of two linear equations shows two straight lines on the same coordinate plane, and their meeting point or overlap tells the solution type.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If two linear equations are given together, what must the final solution satisfy?

Think first, then show answer

The final solution must satisfy both linear equations at the same time. In other words, the same values of x and y should make each equation true together.

Common wrong answer

A common wrong answer is to find only one variable and stop there. That is incomplete because a pair solution needs both x and y.

Don't mix this up

Many students treat the two equations separately and write two different answers. That is wrong because the correct answer must satisfy both equations together.

Exam tip

Always write both equations clearly first, then decide whether graphing, substitution, or elimination will be fastest for the given question.