Two equations, one answer: the graphical method

Two equations together

A boat goes down a river and then back up. We have two pieces of information, and two unknowns. Each piece of information gives one equation. We call two such equations a pair of linear equations in two variables.

A solution of the pair is a value of $x$ and a value of $y$ that make both equations true at the same time.

The graphical method

Each equation is a line. A solution of the pair must be on both lines. So it is the point where the two lines meet.

Example. Solve $x + y = 6$ and $x - y = 2$.

The first line has the points $(6, 0)$, $(0, 6)$ and $(3, 3)$. The second line has the points $(2, 0)$, $(0, -2)$ and $(4, 2)$. We draw both lines on one graph.

The lines cross at the point $(4, 2)$. So $x = 4$ and $y = 2$.

Check. In the first equation, $4 + 2 = 6$. In the second, $4 - 2 = 2$. Both are true.

Make the two slopes equal. The lines turn parallel and the crossing point is gone. Make the two starting values equal as well, and the two lines become one.

Three cases

Two lines in a plane can be in one of three positions.

1. The lines cross at one point. The pair has one solution. We say that the pair is consistent.
2. The lines are parallel. They never meet, so the pair has no solution. We say that it is inconsistent.
3. The lines are the same line. Every point on the line is a solution, so the pair has infinitely many solutions. We say that it is dependent.

Can we tell without drawing?

Yes. Write the pair as

Compare the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$ and $\dfrac{c_1}{c_2}$.

Examples.

• $x + 2y - 4 = 0$ and $2x + 4y - 8 = 0$. The ratios are $\dfrac{1}{2}$, $\dfrac{2}{4}$ and $\dfrac{-4}{-8}$. All three are $\dfrac{1}{2}$. So the lines are the same, and there are infinitely many solutions.
• $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$. The first two ratios are $\dfrac{1}{2}$. The third is $\dfrac{-4}{-12} = \dfrac{1}{3}$. So the lines are parallel, and there is no solution.

Practise with new numbers

Try it in Python

This program tests three pairs of equations with the ratios.

Remember these

• A solution of a pair is a point on both lines.
• One crossing point: one solution. Parallel lines: no solution. The same line: infinitely many solutions.
• The ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, $\dfrac{c_1}{c_2}$ tell the case without drawing.

Check yourself

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

1. Why does a pair whose lines are parallel have no solution?

2. How many solutions does the pair $x + y = 5$ and $2x + 2y = 10$ have?

3. Picture the pair $x = 2$ and $y = 3$. Where do the lines cross?

4. Do the lines $x + 3y = 6$ and $2x + 6y = 15$ meet?