Solving Linear Simultaneous Equations

Slideshow 13, MathematicsMr Richard Sasaki, Room 307

Objectives• Look at multiplying both sides of

an equation by a number• Review the process of solving

simultaneous equations• Solve simultaneous equations for

unknowns with different coefficients

Multiplication WorksheetTry the short multiplication worksheet and fill in the gaps, should take just a few minutes!

Answers4539

11811

-4

2452

(3 𝑥+2)(7 𝑎−3)(9 𝑥+4 )3 𝑦+5

2

3

-2

4 𝑥 – 2=8−2 𝑦+5=−11

Solving Simultaneous EquationsWe can solve simultaneous equations by adding or subtracting when a term and coefficient is the same in both equations.

17 𝑥+5 𝑦=217 𝑥−2 𝑦=23

①②

①－② 7 𝑦=−21𝑦=−3

17 𝑥+5 𝑦=2①

17 𝑥+5(−3)=217 𝑥 – 15=217 𝑥=17𝑥=1

Solving Simultaneous EquationsBut how do we solve simultaneous equations when there is no term and coefficient in common?As in the multiplication worksheet, we can multiply both sides of an equation by a number. We can make it so the same term shares the same coefficient.

Solving Simultaneous EquationsExampleSolve the simultaneous equations below.

What should we do first?The easiest way to make a pair of coefficients the same is to make ②contain . So we will multiply by .②

①

②

②×2

4 𝑥+2 𝑦=6So let’s subtract ①from ×2.②

①

②×2

Let’s substitute into .②2 𝑥+𝑦=3②

2(2)+𝑦=34+𝑦=3𝑦=−1

So and .

Solving Simultaneous EquationsExampleSolve the simultaneous equations below.

①

②

Let’s multiply ① by 3 and by 2. Then we will ②have in both equations.6 𝑎 –15𝑏=33① x 3

② x 26 𝑎+4𝑏=14① x 3 - ② x 2−19𝑏=19

𝑏=−1①2𝑎 – 5(−1)=112𝑎+5=112𝑎=6𝑎=3

So and .

Try the worksheets!

Answers

𝑥=6 , 𝑦=1 𝑎=3 ,𝑏=6

𝑥=2 , 𝑦=2𝑥=2 , 𝑦=−1

𝑥=5 , 𝑦=6𝑥=10 , 𝑦=1𝑎=½ ,𝑏=3𝑥=4 , 𝑦=−1𝑥=−2 , 𝑦=1𝑥=2 , 𝑦=−3

𝑥=3 , 𝑦=8 𝑥=1 , 𝑦=4

𝑥=2 , 𝑦=−1 𝑥=4 , 𝑦=3

𝑥=−2 , 𝑦=−3 𝑥=2 , 𝑦=9

𝑥=−3 , 𝑦=3𝑥=10 , 𝑦=15

𝑥=−5 , 𝑦=13𝑥=−17 , 𝑦=20

Answers

𝑚=5½ ,𝑛=1𝑥=3 , 𝑦=4

𝑥=4 , 𝑦=3 𝑥=1 , 𝑦=1

𝑎=9½ ,𝑏=5½Chicken eggs: 20gDuck eggs: 35g

There are 15 two yen coins and 25 five yen coins