# Solve Simultaneous Equations By Elimination

The purpose of solving simultaneous equations is to find the **same x-value** and the **same y-value** that satisfies both equations.

To solve, the **like terms of two equations are lined up under each other**, the whole equations are added or subtracted, and **one variable is eliminated** in the process.

When adding numbers, we line up the digits in columns of hundreds, tens and units...

When adding whole equations, we line up the like or similar terms, but a variable is eliminated.

**This is how the process starts...**

3x + 2y + 5 = 0 |

+ (4x – 2y + 9 = 0) |

7x + 14 = 0 |

**Here's a demonstration...**

## Example One - Elimination by Adding

Solve these equations by elimination:

4x + y = 9

6x – y = 11

**Answer:**

4x + y = 9 |

+ (6x – y = 11) |

10x = 20 |

10x = 20

x = 2

*Now, we substitute the x value into one of the equations to find the y value...*

4x + y = 9

8 + y = 9

y = 1

*The simultaneous solution is x = 2 and y = 1.*

## Example Two - Elimination by Subtracting

Solve these equations by elimination:

5y + 7x = 8

3y + 7x = 2

**Answer:**

*Remember that subtracting the second equation means that all signs of the second equation change to their opposite signs.*

5y + 7x = 8 |

– (3y + 7x = 2) |

2y = 6 |

2y = 6

y = 3

*Now, we substitute the y value into one of the equations to find the x value...*

5y + 7x = 8

15 + 7x = 8

7x = 8 – 15

7x = –7

x = –1

*The simultaneous solution is x = –1 and y = 3.*

## Example Three - Lining Up Whole Equations First

Solve these equations by elimination:

4x – 3y = 26

x – 11 = 3y

**Answer:**

4x – 3y = 26 |

– (x – 3y = 11) |

3x = 15 |

3x = 15

x = 5

Now, we substitute the x value into one of the equations to find the y value...

x – 11 = 3y

5 – 11 = 3y

–6 = 3y

–2 = y

y = –2

*The simultaneous solution is x = 5 and y = –2.*

## Example Four - Multiplying Whole Equations First

Solve these equations by elimination:

4x + 3y = 11

3x – 2y = 21

**Answer:**

*I intend to eliminate the y variable.
*

I will multiply the first equation by 2.

I will multiply the second equation by 3.

Then there will be 6y in both equations...

(4x + 3y = 11) × 2

(3x – 2y = 21) × 3

8x + 6y = 22

9x – 6y = 63

*Now, I will add these equations to eliminate y...*

8x + 6y = 22 |

+ (9x – 6y = 63) |

17x = 85 |

17x = 85

x = 5

*Now, we substitute the x value into any equation to find the y value...*

4x + 3y = 11

20 + 3y = 11

3y = 11 – 20

3y = –9

y = –3

*The simultaneous solution is x = 5 and y = –3.*

## Questions - Solve by Elimination

**Q1.**

x + y = 6

x – y = 4

**Q2.**

3x + 5y = 14

2x – 5y – 1 = 0

**Q3.**

x – 2y = –5

2x + 3y – 4 = 0

**Answers**

**A1.** x = 5, y = 1

**A2.** x = 3, y = 1

**A3.** x = –1, y = 2