Solving by substitution

Why not only draw?

A graph is good for seeing, but it is not exact. If the answer is $x = 2.8$, we can only guess it from a drawing. We want a method that gives the exact answer. The substitution method is one.

The idea

From one equation, we get $y$ in terms of $x$. Then we put this in the other equation. The other equation now has only $x$ in it, and we can solve it.

The steps

1. Take one equation. Write $y$ in terms of $x$ (or $x$ in terms of $y$).
2. Substitute this in the other equation. Now there is only one variable.
3. Solve this equation.
4. Put the answer back in the first step to find the other variable.
5. Check the answer in both equations.

Example 1

Solve $2x + y = 11$ and $x - y = 1$.

Step 1. The second equation is simple. We want $x$ in terms of $y$, so $x = y + 1$.

Step 2. We put $x = y + 1$ in the first equation.

Step 3. We solve it. $2y + 2 + y = 11$, so $3y = 9$, and $y = 3$.

Step 4. We find $x$. $x = y + 1 = 3 + 1 = 4$.

Step 5. Check. $2 \times 4 + 3 = 11$ is true. $4 - 3 = 1$ is true.

The solution is $x = 4$ and $y = 3$.

Example 2

Solve $x + 2y = 11$ and $3x - y = 12$.

The second equation has $y$ with a simple number in front. We want $y$, so $y = 3x - 12$.

We substitute it in the first equation: $x + 2(3x - 12) = 11$.

We solve it: $x + 6x - 24 = 11$, so $7x = 35$, and $x = 5$. Then $y = 3 \times 5 - 12 = 3$.

Check. $5 + 2 \times 3 = 11$ is true. $3 \times 5 - 3 = 12$ is true.

The solution is $x = 5$ and $y = 3$.

Change $s$ and watch the crossing move. The answer of the one-variable equation always lands right under the crossing.

When the pair has no solution

Try $x + y = 2$ and $2x + 2y = 6$. From the first, $y = 2 - x$. We put it in the second: $2x + 2(2 - x) = 6$. This gives $2x + 4 - 2x = 6$, which is $4 = 6$. This is never true. This tells us that the pair has no solution. The lines are parallel.

If we got $6 = 6$ (always true), the pair would have infinitely many solutions.

Practise with new numbers

Try it in Python

Python can solve a pair at once. This program solves $x + 2y = 11$ and $3x - y = 12$, and it checks both equations.

Remember these

• Substitution: write one variable in terms of the other, put it in the second equation, and solve.
• Always check in both equations.
• If you get a false statement like $4 = 6$, there is no solution. If you get a true statement like $6 = 6$, there are infinitely many.

Check yourself

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

1. Solve $x + y = 10$ and $x - y = 2$ by substitution.

2. Solve $y = 2x$ and $3x + y = 15$.

3. You get $0 = 5$ while solving a pair. What does it tell you?

4. Solve $x = 3y$ and $x + y = 16$.