Solving by elimination, and problems in words

The elimination method

In the elimination method, we add or subtract the two equations so that one variable disappears. This is quick when the numbers in front of one variable are the same, or opposite.

Example 1: the numbers are opposite

Solve $3x + 2y = 12$ and $5x - 2y = 4$.

The numbers in front of $y$ are $+2$ and $-2$. They are opposite. If we add the equations, $y$ disappears.

We put $x = 2$ in the first equation: $6 + 2y = 12$, so $y = 3$.

Check. $3 \times 2 + 2 \times 3 = 12$ is true. $5 \times 2 - 2 \times 3 = 4$ is true.

The solution is $x = 2$ and $y = 3$.

Example 2: we first make the numbers equal

Solve $2x + 3y = 13$ and $5x - 2y = 4$.

No variable has the same number in both equations. So we multiply the equations to make the numbers in front of $y$ equal in size. We multiply the first equation by 2, and the second by 3.

Now the numbers in front of $y$ are $+6$ and $-6$. We add the equations: $19x = 38$, so $x = 2$.

We put $x = 2$ in the first equation: $4 + 3y = 13$, so $y = 3$.

Check. $2 \times 2 + 3 \times 3 = 13$ is true. $5 \times 2 - 2 \times 3 = 4$ is true.

The solution is $x = 2$ and $y = 3$.

Move $k$ to $1$. The green line stands straight up at $x = 2$, because adding the two equations removes $y$. That is the whole idea of elimination.

Which method should I use?

• If one equation already shows $y = \ldots$ or $x = \ldots$, use substitution.
• If the numbers in front of one variable are equal or opposite, use elimination.
• Use a graph when you want to see the situation, or to check.

All methods give the same answer. If they do not, there is a mistake.

Problems in words

Many problems are given in words. Follow these steps.

1. Read the problem. Decide what the two unknowns are, and name them $x$ and $y$.
2. Write two equations from the two facts in the problem.
3. Solve the pair.
4. Check the answer in the words of the problem, not only in the equations.

Example. 2 notebooks and 3 pens cost 17 dollars. 3 notebooks and 2 pens cost 18 dollars. Find the price of a notebook and of a pen.

Let a notebook cost $x$ dollars and a pen cost $y$ dollars. Then

We multiply the first by 3 and the second by 2: $6x + 9y = 51$ and $6x + 4y = 36$. We subtract: $5y = 15$, so $y = 3$. Then $2x + 9 = 17$, so $x = 4$.

A notebook costs 4 dollars, and a pen costs 3 dollars.

Check in words. 2 notebooks and 3 pens: $8 + 9 = 17$ dollars. Yes. 3 notebooks and 2 pens: $12 + 6 = 18$ dollars. Yes.

Practise with new numbers

Try it in Python

This program solves a river problem. A boat goes 24 km downstream in 2 hours and 24 km upstream in 3 hours. It finds the speed of the boat in still water and the speed of the stream.

Remember these

• Elimination: add or subtract the equations so that one variable disappears. Multiply first if you need to.
• For a problem in words, name the unknowns, write two equations, solve, and check in the words.
• All methods must give the same answer.

Check yourself

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

1. Solve $x + y = 9$ and $x - y = 3$ by elimination.

2. The sum of two numbers is $30$. Their difference is $8$. Find the numbers.

3. Why do we check the answer in the words of a problem?

4. Solve $2x + 3y = 16$ and $2x + y = 8$ by elimination.