A pair of equations and its graph

Two conditions at once

Taxi A charges ₹40 to start and ₹12 for each kilometre. Taxi B charges ₹70 to start and ₹9 for each kilometre. After how many kilometres do the two taxis cost the same?

Let $x$ be the distance in kilometres and $y$ the cost in rupees. Each taxi gives one equation:

Moving every term to the left puts them in standard form:

A linear equation in two variables has the form $ax + by + c = 0$, where $a$, $b$ and $c$ are numbers and $a$ and $b$ are not both zero. Two such equations taken together are a pair of linear equations in two variables:

A solution of the pair is a pair of numbers $(x, y)$ that makes both equations true at the same time.

One equation has many solutions

Take $x + y = 5$. Choose a value of $x$ and work out $y$. This makes a table of values.

Plot these points. They lie on one straight line. The graph of a linear equation in two variables is a straight line, every point on it is a solution, and every solution is a point on it.

Two points are enough to draw a line. A third point is a check: if it is not on the line, one of the points is wrong.

The solution of a pair

Now draw $x + y = 5$ and $2x - y = 1$ on the same grid. For the second line, $x = 0, 1, 2, 3$ give $y = -1, 1, 3, 5$.

The lines cross at $(2, 3)$. This point is on both lines, so it fits both equations:

Any other point is on at most one of the lines, so $(2, 3)$ is the only solution of the pair.

A drawing is only as exact as your pencil. Always check a solution read from a graph by putting the numbers into both equations.

The taxi problem on a graph

With the sliders at the start, the lines meet at $(10, 160)$. After $10$ km both taxis cost ₹160. For a shorter ride company A is cheaper, because it has the lower starting fee. For a longer ride company B is cheaper, because its rate for each kilometre is lower.

Make the two rates equal and the lines run side by side: they never meet, so the costs are never equal. This idea returns in the next lesson.

Practise with new numbers

Try it in Python

A computer can tabulate the two fares and search for the distance where they are equal.

Common mistakes

• Drawing a line from one point. Plot at least two points, and a third as a check.
• Swapping the axes. The point $(2, 3)$ has $x = 2$ and $y = 3$, in that order.
• Checking only one equation. A solution of a pair must fit both equations.
• Trusting a rough graph. Read the point, then check it by putting the numbers into both equations.

Remember these

• A linear equation in two variables has a straight line as its graph. Every point on the line is a solution.
• A pair of equations is $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
• A solution of the pair is a point on both lines, and it fits both equations.
• Always check a solution in both equations.

Check yourself

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

1. Is $(3, 1)$ a solution of $x + 2y = 5$?

2. On the line $3x + y = 14$, find $y$ when $x = 4$.

3. Two lines cross at $(6, 2)$. What is the solution of the pair?

4. How many points do you need to draw a straight line, and why use one more?

5. The pair has the solution $(1, 4)$. Does $(1, 4)$ fit the equation $2x - y = -2$?