Linear simultaneous equations

In a nutshell

Simultaneous equations with two unknowns will have one set of values that will make a pair of equations true. Simultaneous equations can be worked out algebraically or by using your calculator.

Elimination method

To solve a pair of simultaneous equations, you can use the elimination method. To do this, you eliminate one of the unknown variables, and calculate the value of the other variable. By substituting this value into one of the original equations, you can work out the remaining unknown variable. Alternatively, you can use your calculator which will work out the variables for you, however it is important to know how to work algebraically in case you cannot use your calculator.

Example 1

Solve the simultaneous equations:

$$\begin{aligned}7x+3y&=16\\2x+9y&=29\end{aligned}$$​

Multiply the first equation by $$3$$ to get $$9y$$ in both equations:

$$\begin{aligned}21x+9y&=48\\2x+9y&=29\end{aligned}$$​​

Subtract the second equation from the first and solve for $$x$$​:

$$\begin{aligned}19x&=19\\x&=1\end{aligned}$$​​​​

Substitute this value for $$x$$ into one of the equations and solve for $$y$$:

$$\begin{aligned}7(1)+3y&=16\\3y&=9\\y&=3\end{aligned}$$

Therefore: 

$$\underline{x=1,\ y=3}$$​​

Substitution method

To use the substitution method to solve a pair of simultaneous equations, you first rearrange one of the equations to make one variable the subject. You then substitute the expression for this variable into the other equation to make one linear equation, and solve. By substituting the value for this variable into one of the original equations or the expression you created, you can work out the other variable.

Example 2

Solve the simultaneous equations:

$$\begin{aligned}5x+2y&=6\\3x-10y&=26\end{aligned}$$​​

Rearrange the first equation to make $$y$$ the subject:

$$\begin{aligned}2y&=6-5x\\y&=3-\dfrac52x\end{aligned}$$​​

Substitute this into the second equation and solve for $$x$$​:

$$\begin{aligned}3x-10(3-\dfrac52x)&=26\\3x-30+25x&=26\\28x&=56\\x&=2\end{aligned}$$​​

Substitute this value for $$x$$ into the equation for $$y$$:

$$\begin{aligned}y&=3-\dfrac52(2)\\y&=3-5\\y&=-2\end{aligned}$$​​

The solutions to the simultaneous equations are $$\underline{x=2,\ y=-2}$$.​

​​​

Solve simultaneous equations using a calculator

To solve simultaneous equations on a calculator, use equations/functions from the menu. Press "simul equation" and select $$2$$ for the number of unknowns (the two unknowns are $$x$$ and $$y$$). Then input the equations as they appear and press equals for the value of $$x$$. Press equals again to get the value for $$y$$.​

CALCULATOR TIP

​

$$\boxed{\begin{aligned}1:Simul\ Equation\\Number\ of\ \ \ \ \ \ \ \ \\\ \ \ \ \ Unkowns?\\Select\ 2\sim 4\ \ \ \ \ \ \ \end {aligned}}$$​​

$$\boxed{\begin{aligned}0x+\ \ \ \ \ \ 0y\ \ \ \ \ \ = 0\\0x+\ \ \ \ \ \ 0y\ \ \ \ \ \ = 0\end {aligned}}$$​​

​

Example 3

Solve the simultaneous equations: 

$$\begin{aligned}2x+3y&=8\\3x -y&=23\end{aligned}$$​​

To calculate $$x$$ and $$y$$ use the calculator with $$2$$ unknown variables and input the equations:

	​$$\boxed{\begin{aligned}2x+\ \ \ \ \ \ 3y&=\ \ \ \ \ \ 8\\3x -\ \ \ \ \ \ \ \ y&=\ \ \ \ \ \ 23\end {aligned}}$$​​
	​$$\boxed{x=7,\ y=-2}$$​​

Therefore, $$\underline{x=7,\ y=-2}$$.