Home

Maths

Algebra

Simultaneous equations: elimination and substitution

Simultaneous equations: elimination and substitution

Select Lesson

Exam Board

Select an option

Explainer Video

Loading...
Tutor: Bilal

Summary

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. 


PROCEDURE

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

Subtract

Different sign

Add

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.


PROCEDURE

1.1.​​

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

2.2.​​

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

3.3.​​

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

4.4.​​

Check both answers in equation 22.​


Example 2

Solve

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}


​​



Create an account to read the summary

Exercises

Create an account to complete the exercises

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?

Beta

I'm Vulpy, your AI study buddy! Let's study together.