Simultaneous equations: elimination and substitution

Simultaneous equations: elimination and substitution

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Simultaneous equations: elimination and substitution

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 or the substitution method.

Elimination method

The elimination method involves writing the equations in the correct form and in columns so that the equations can be added or subtracted to eliminate a variable. 


1. 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.

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

Same sign


Different sign


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

4. Check both answers in the other equation.

Example 1

7c+2d=217c5d=1623d=182+1d=67c+2×6=27c=14c=27×25×6=16c=2,d=6\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}​​

Substitution method

The substitution method involves rearranging one of the equations and substituting into the other equation in order to find the solutions.



Rearrange one of the equations for either xx or yy. Call this equation 11 and call the other equation 22


Substitute equation 11​ into equation 22​ and solve for the unknown variable.


Substitute the answer from step 22 into equation 11 and solve.​


Check both answers in equation 22.​

Example 2


xy=32x+y=3\begin{aligned}x-y&=3 \\2x+y&=3 \\\end{aligned}​​

Rearrange the first equation for xx and label the equations.

x=y+312x+y=32\begin{aligned}x& = y+3 \qquad &\textcircled{1} \\2x+y &=3 \qquad &\textcircled{2} \\ \\\end {aligned}​​

Substitute equation 11 into equation 22 and solve.

2(y+3)+y=32y+6+y=33y+6=33y=3y=1\begin{aligned}2(y+3)+y &=3 \\2y+6+y &= 3 \\3y + 6 &= 3 \\3y &=-3 \\y &= -1\end {aligned}​​

Substitute the y=1y=-1 into equation 11.

x=y+3x=1+3x=2\begin{aligned}x &=y+3 \\x &=-1+3 \\x &= 2\end{aligned}​​

Check the answers in equation 22.

2x+y=32×21=3\begin{aligned}2x+y &= 3 \\2 \times 2 - 1 &= 3 \\\end {aligned}​​

x=2,y=1\underline{x = 2, y=-1}


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FAQs - Frequently Asked Questions

What is the elimination method?

What are the rules of simultaneous equations when using the elimination method?

What are the methods for solving simultaneous equations?


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