Mathematics » Equations and Inequalities » Solving Simultaneous Equations

# Solving by Substitution

## Solving by Substitution

• Use the simplest of the two given equations to express one of the variables in terms of the other.

• Substitute into the second equation. By doing this we reduce the number of equations and the number of variables by one.

• We now have one equation with one unknown variable which can be solved.

• Use the solution to substitute back into the first equation to find the value of the other unknown variable.

The following video shows how to solve simultaneous equations using substitution.

## Example

### Question

Solve for $$x$$ and $$y$$:

\begin{align*} x – y & =1 \qquad \ldots(1) \\ 3 & = y – 2x \qquad \ldots(2) \end{align*}

### Use equation $$(1)$$ to express $$x$$ in terms of $$y$$

$x=y+1$

### Substitute $$x$$ into equation $$(2)$$ and solve for $$y$$

\begin{align*} 3 & = y – 2(y + 1) \\ 3 & = y – 2y – 2 \\ 5 & = -y \\ \therefore y & = -5 \end{align*}

### Substitute $$y$$ back into equation $$(1)$$ and solve for $$x$$

\begin{align*} x & = (-5) + 1 \\ \therefore x & = -4 \end{align*}

### Check the solution by substituting the answers back into both original equations

\begin{align*} x & = -4 \\ y & = -5 \end{align*}

## Example

### Question

Solve the following system of equations:

\begin{align*} 4y + 3x & = 100 \qquad \ldots(1) \\ 4y – 19x & = 12 \qquad \ldots (2) \end{align*}

### Use either equation to express $$x$$ in terms of $$y$$

\begin{align*} 4y + 3x & = 100 \\ 3x & = 100 – 4y \\ x & = \cfrac{100 – 4y}{3} \end{align*}

### Substitute $$x$$ into equation $$(2)$$ and solve for $$y$$

\begin{align*} 4y – 19\left(\cfrac{100 – 4y}{3}\right) & = 12 \\ 12y – 19(100 – 4y) & = 36 \\ 12y – \text{1 900} + 76y & = 36 \\ 88y & = \text{1 936} \\ \therefore y & =22 \end{align*}

### Substitute $$y$$ back into equation $$(1)$$ and solve for $$x$$

\begin{align*} x & = \cfrac{100 – 4(22)}{3} \\ & = \cfrac{100 – 88}{3} \\ & = \cfrac{12}{3} \\ \therefore x & = 4 \end{align*}