Substitution method

The substitution method consists in isolating one of the unknowns in one equation and then inserting this expression into the other equation. In this way we obtain a new equation with only one unknown, which we can solve. Once the first unknown has been found, we can then find the other.

 

The method can be used for all systems of equations with two unknowns, but it can be a bit more cumbersome if the numbers are large or awkward.

 

We have a system of equations with two equations:

 

$$ \large 8y-4x=4 $$

$$ \large 2y+4x=20 $$

 

Here we cannot directly read off the value of \( \large x\) and \( \large y\), so we need to find it first.

We take the top equation and isolate \( \large x\):

 

$$ \large \begin{aligned} 8y-4x &=4 \quad \Leftrightarrow \\[12pt]  8y &=4+4x \quad \Leftrightarrow \\[12pt]  8y-4 &=4x \quad \Leftrightarrow \\[12pt] \frac{8y-4}{4} &=\frac{4x}{4} \quad \Leftrightarrow \\[12pt] 2y-1 &=x \end{aligned}$$

 

We now have an expression for the value of \( \large x\)

 

 

Find the first unknown

 

$$ \large 8y-4x=4 $$

$$ \large 2y+4x=20 $$

 

We now insert the value of \(\large x\) into equation 2

 

$$ \large 2y+4\color{red}{(2y-1)}=20 $$

 

We need to remove the parentheses by expanding:

 

$$ \large \begin{aligned} 2y+8y-4&=20 \quad \Leftrightarrow \\[12pt]  10y-4&=20 \quad \Leftrightarrow \\[12pt]  \frac{10y-4}{10}&=\frac{20}{10} \quad \Leftrightarrow \\[12pt]  y-\frac{4}{10} &= 2 \quad \Leftrightarrow \\[12pt]  y&= 2+ \frac{4}{10} \quad \Leftrightarrow \\[12pt]  y &= 2\frac{4}{10} \quad \Leftrightarrow  \\[12pt]  y &= 2.4  \end{aligned} $$

 

 

Find the second unknown

We found earlier that:

 

$$ \large x=2y-1 $$

 

We found that:

 

$$ \large y = 2.4 $$

 

So now we can insert \(\large y\) into the equation for \(\large x\):

 

$$ \large x=2y-1 \quad \Leftrightarrow   $$

$$ \large x=2\cdot 2.4 -1 \quad \Leftrightarrow   $$

$$ \large x=4.8 -1 \quad \Leftrightarrow   $$

$$ \large x=3.8 $$

 

Check:

 

$$ \large 8\cdot2.4-4\cdot3.8=4 $$

$$ \large 2\cdot2.4+4\cdot3.8=20 $$

 

Both are true!

 

 

Note: The substitution method always works, but sometimes it shows that the system has no solution or infinitely many:

 

• If you end up with something impossible, e.g. \( \large 0 = 5 \), it means the system has no solution.
• If you end up with something trivial, e.g. \( \large 0 = 0 \), it means the system has infinitely many solutions.