Solving a system by substitution

The substitution method turns the system into an equation in one unknown – one you already know how to solve. The steps:

• Isolate an unknown in one of the equations, for example $\displaystyle {y}={\ldots}$
• Substitute the expression you got for that unknown in the other equation – you get an equation in one unknown.
• Solve it.
• Put the result into the isolated expression, and find the other unknown.
• Write the solution as an ordered pair $\displaystyle (x,y)$ and check it in both equations.

On a graph the meeting point can be read exactly only when it is on a whole square. Substitution finds it by calculation, wherever it is.

The solution below substitutes $\displaystyle 2x+1$ for $\displaystyle y$ in the second equation:

• $\displaystyle y$ comes from the isolated equation: $\displaystyle {y}={2\cdot2+1}={5}$.
• The solution of the system: $\displaystyle (2,5)$.
• Check: $\displaystyle {5}={2\cdot2+1}$ and $\displaystyle {3\cdot2+5}={11}$.

Here no unknown is isolated. The solution below isolates $\displaystyle x$ in the first equation, $\displaystyle {x}={8-2y}$, and substitutes it in the second:

• $\displaystyle {x}={8-2\cdot3}={2}$.
• The solution of the system: $\displaystyle (2,3)$.
• Check: $\displaystyle {2+2\cdot3}={8}$ and $\displaystyle {3\cdot2-3}={3}$.

The expression you substitute goes in brackets. A minus before the brackets changes the sign of every term inside.
In the system $\displaystyle {y}={x-3}$, $\displaystyle {4x-y}={12}$:

Then $\displaystyle {y}={3-3}={0}$, and the solution is $\displaystyle (3,0)$. Without the brackets we would write $\displaystyle 4x-x-3$ – and get the sign of 3 wrong.

$\displaystyle {\begin{cases}y=x+3\\ 2x+y=12\end{cases}}$
Substituting $\displaystyle x+3$ for $\displaystyle y$ in the second equation gives the equation below, with $\displaystyle x$ alone.
Solve it.
Work out $\displaystyle y$.
Write the solution of the system and check it in both equations.

Solve by substitution.