# Simultaneous equations

## In a nutshell

Simultaneous equations involves solving two (or more) equations, where the answers work in all equations given. The idea is that there is a set of solutions that work in both the equations. For two unknowns, there needs to be two equations to solve. Simultaneous equations can be solved using the elimination method.

#### Procedure

A) Check if the equations need to be rearranged or multiplied up for them to be written in the correct format. Write out the equations, and number them.

B) Decide whether to add or subtract equations, and solve for one of the variables. Use the rule:

**S**ame sign | **S**ubtract |

**D**ifferent sign | Ad**d** |

C) Substitute the answer obtained into one of the equations, and solve for the other variable.

D) Check both answers in the other equation.

##### Example 1

$\begin {aligned}7c+5t &=29 &\textcircled{1} \\7c+8t&= 38 &\textcircled{2} \\\underline{\qquad \quad}& \underline{\qquad \qquad \quad} \\3t&=9 &\textcircled{2} - \textcircled{1}\\t&=3 \\\\7c +5\times3 &=29 \\7c &=14 \\c&=2 \\\\7\times2 + 8\times 3 &=38 \\\underline {c=2, t=3}\end {aligned}$

##### Example 2

$\begin {aligned}7c+2d &=2 &\textcircled{1} \\-7c-5d&= 16 &\textcircled{2} \\\underline{\qquad \quad}& \underline{\qquad \qquad \quad} \\-3d&=18 &\textcircled{2} + \textcircled{1}\\d&=-6 \\\\7c +2\times-6 &=2 \\7c &=14 \\c&=2 \\\\-7\times2 -5\times -6 &=16 \\\underline {c=2, d=-6}\end {aligned}$

##### Example 3

$\begin {aligned}5p+6q&=17 \\2p+3q&=5 \qquad &\times2\\\\5p+6q&=17 \qquad &\textcircled{1} \\4p+6q&=10 \qquad &\textcircled{2} \\\underline{\qquad \quad}& \underline{\qquad \qquad \quad} \\p&=7 \qquad &\textcircled{1} - \textcircled{2}\\\\5\times7 +6q &=17 \\6q &=-18 \\q&=-3 \\\\2\times7 +3\times -3 &=5 \\\underline {p=7,q=-3}\end {aligned}$