# Simultaneous Equations

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## Learning Checklist

Understanding Simultaneous Equations

Examples

## Linear Simultaneous Equations

**What are linear simultaneous equations?**

- When there are two unknowns (sayÂ
) in a problem, we need two equations to be able to find them both: these are called simultaneous equations*x*Â andÂ*y*- you solveÂ
**twoÂ**equations to findÂ**twoÂ**unknowns,Â*x*Â andÂ*y*- for example, 3
*x*Â + 2*y*Â = 11 and 2*x*Â –Â*y*Â = 5 - the solutions areÂ
*x*Â = 3 andÂ*y*Â = 1

- for example, 3

- you solveÂ
- If they just have
*Â x*Â andÂ*y*Â in them (noÂ*x*^{2}Â orÂ*y*^{2}Â orÂ*xy*Â etc) then they areÂ**linear**Â simultaneous equations

**How do we solve linear simultaneous equations by elimination method?**

- “Elimination”Â completely removes one of the variables,Â
*xÂ*orÂ*y* - To eliminate theÂ
*x*‘s from 3*x*Â + 2*y*Â = 11 and 2*x*Â –Â*y*Â = 5- Multiply every term in the first equation by 2
- 6
*x*Â + 4*y*Â = 22

- 6
- Multiply every term in the second equation by 3
- 6
*x*Â – 3*y*Â = 15

- 6
**Subtract**Â the second result from the first to eliminate the 6x’s, leaving 4*y*Â – (-3*y*) = 22 – 15, i.e.Â 7*y*Â = 7- Solve to findÂ
*y*Â (*y*Â = 1) then substituteÂ*y*Â = 1 back into either original equation to findÂ*x*Â (*x*Â = 3)

- Multiply every term in the first equation by 2
- Alternatively, to eliminate theÂ
*y*‘s from 3*x*Â + 2*y*Â = 11 and 2*x*Â –Â*y*Â = 5- Multiply every term in the second equation by 2
- 4
*x*Â – 2*y*Â = 10

- 4
**Add**Â this result to the first equation to eliminate the 2*y*‘s (as 2*y*Â + (-2*y*) = 0)- The process then continues as above

- Multiply every term in the second equation by 2
**Check**Â your final solutions satisfy both equations

**How do we solve linear simultaneous equations by the substitution method?**

- “Substitution” means substituting one equation into the other
- Solve 3
*x*Â + 2*y*Â = 11 and 2*x*Â –Â*y*Â = 5 by substitution- Rearrange one of the equation intoÂ
*y*Â = … (orÂ*x*Â = …)- For example, the second equation becomesÂ
*y*Â = 2*x*Â – 5Â

- For example, the second equation becomesÂ
- Substitute this into the first equation (replace allÂ
*y*‘s with 2*x*Â – 5 in brackets)- 3
*x*Â + 2(2*x*Â – 5)Â = 11

- 3
- Solve this equation to findÂ
*x*Â (*x*Â = 3), then substituteÂ*x*Â = 3 intoÂ*y*Â = 2*x*Â – 5 to findÂ*y*Â (*y*Â = 1)

- Rearrange one of the equation intoÂ
**Check**Â your final solutions satisfy both equations

**How do we solve linear simultaneous equations by graphical method?**

**Plot**Â both equations on the same set of axes- to do this, you can use a table of values or rearrange it into
*y*Â =Â*mx*Â +Â*cÂ*if that helps.

- to do this, you can use a table of values or rearrange it into
- Find where the linesÂ
**intersect**Â (cross over)- TheÂ
*x*Â andÂ*yÂ***solutions**Â to the simultaneous equations are theÂ*xÂ*andÂ*yÂ***coordinates**Â of the point ofÂ**intersection**

- TheÂ
- e.g. to solve 2
*xÂ*–Â*y*Â = 3 and 3*x*Â +Â*y*Â = 4 simultaneously, first plot them both (see graph)- find the point of intersection, (2, 1)
- the solution isÂ
*x*Â = 2 andÂ*y*Â =Â

## Examples

#### Solve the simultaneous equations

5*x*Â + 2*y*Â = 11

4*x*Â – 3*y*Â = 18

Number the equations.

5*x*Â + 2*y* = 11Â Â Â ——->1

4*x*Â – 3*y* = 18Â Â Â ——->2

Make theÂ *y*Â terms equal by multiplying all parts of equation (1) by 3 and all parts of equation (2) by 2.

This will give two 6*yÂ *terms with different signs.Â The question could also be done by making theÂ *xÂ *terms equal by multiplying all parts of equation (1) by 4 and all parts of equation (2) by 5, and subtracting the equations.

15*x*Â + 6*y* = 33Â Â Â ——->3Â Â Â

4*x*Â – 3*y* = 18Â Â Â Â ——->4

The 6*y*Â terms have different signs, so they can be eliminated by adding equation (4) to equation (3).Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 15*x*Â + 6*y* = 33Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â +(4*x*Â – 3*y* = 18)Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â _______________

19x=

## Examples:

## Let's Summarize

## Test Your Understanding

## Olympiad Level Questions

Practice Quiz(download **meandmath practice app**)

## Still Stuck!

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## Related Topics

- Understanding Whole as a Percentage
- Simplifying Percentages
- FDP
- Finding 10%
- Finding 5%
- Finding % of an amount
- Percentage Increase & Decrease

## Learning Checklist

- Understanding Percentages
- Percentage symbol
- Understanding percentages visually

## Understanding Percentage

A percentage is a number that is expressed as a part of 100.

**Per** means **Out Of**, & **cent** means **100**.

Let’s discuss more.

5% means 5 out of 100.

12% means 12 out of 100.

## Let's learn through visuals.

## Examples

A percentage can be represented by shading in a 100 square grid.

## Let's Summarize

### Percentage

AÂ percentageÂ is a number which is expressed as aÂ **partÂ of 100**.Â Â

**Per**Â meansÂ **out of,**Â &Â **cent**Â meansÂ **100**.

Symbol: %

## Test Your Understanding

## Olympiad Level Questions

Practice Quiz(download **meandmath practice app**)

## Still Stuck!

Book a free demo class & clear your doubts!

## Related Topics

- Understanding Whole as a Percentage
- Simplifying Percentages
- FDP
- Finding 10%
- Finding 5%
- Finding % of an amount
- Percentage Increase & Decrease