Slideshow 13, Mathematics
Mr Richard Sasaki, Room 307
• 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
Try the short multiplication worksheet and
fill in the gaps, should take just a few
minutes!
11
8
11
4
5
3
9
2
-4
(3𝑥 + 2)
3
2
4
5
2
(7𝑎 − 3)
(9𝑥 + 4)
3𝑦 + 5
-2
4𝑥 – 2 = 8
−2𝑦 + 5 = −11
We can solve
simultaneous
equations by adding
or subtracting when a
term and coefficient is
the same in both
equations.
① 17𝑥 + 5𝑦 = 2
② 17𝑥 − 2𝑦 = 23
①－
②
7𝑦 = −21
𝑦 = −3
①17𝑥 + 5𝑦 = 2
17𝑥 + 5(−3) = 2
17𝑥 – 15 = 2
17𝑥 = 17
𝑥 = 1
But 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.
Example
Solve the simultaneous equations below.
What should we do first?
So let’s subtract ① ① 3𝑥 + 2𝑦 = 4
② 2𝑥 + 𝑦 = 3 The easiest way to make a
from ②×2.
pair of coefficients the same
②×2 4𝑥 + 2𝑦 = 6 ②×2
4𝑥 + 2𝑦 = 6is to make ② contain 2𝑦. So
①
3𝑥 + 2𝑦 = 4
𝑥
= 2
Let’s substitute 𝑥 = 2 into ②.
we will multiply ② by 2.
2𝑥 + 𝑦 = 3
2(2) + 𝑦 = 3
So 𝑥 = 2 and 𝑦 = −1.
4+ 𝑦 = 3
𝑦 = −1
②
Example
Solve the simultaneous equations below.
2𝑎 – 5𝑏 = 11
② 3𝑎 + 2𝑏 = 7
Let’s multiply ① by 3 and ② by 2. Then we will have 6𝑎
in both equations.
① x 3 6𝑎 – 15𝑏 = 33
①
② x 26𝑎 + 4𝑏
① x 3 - ② x 2−19𝑏
= 14
= 19
𝑏 = −1 So 𝑎 = 3 and 𝑏 = −1.
①2𝑎 – 5(−1) = 11 Try the worksheets!
2𝑎 + 5 = 11
2𝑎 = 6
𝑎 = 3
𝑥 = 6, 𝑦 = 1 𝑎 = 3, 𝑏 = 6
𝑥 = 3, 𝑦 = 8
𝑥 = 1, 𝑦 = 4
𝑥 = 2, 𝑦 = 2 𝑥 = 2, 𝑦 = −1
𝑥 = 2, 𝑦 = −1
𝑥 = 4, 𝑦 = 3
𝑥 = 5, 𝑦 = 6
𝑥 = 10, 𝑦 = 1
𝑎 = ½ , 𝑏 = 3 𝑥 = 4, 𝑦 = −1
𝑥 = −2, 𝑦 = 1
𝑥 = 2, 𝑦 = −3
𝑥 = −2, 𝑦 = −3 𝑥 = 2, 𝑦 = 9
𝑥 = −3, 𝑦 = 3 𝑥 = 10, 𝑦 = 15
𝑥 = −5, 𝑦 = 13
𝑥 = −17, 𝑦 = 20
𝑚 = 5½, 𝑛 = 1 𝑥 = 3, 𝑦 = 4
𝑥 = 4, 𝑦 = 3
𝑥 = 1, 𝑦 = 1
𝑎 = 9½, 𝑏 = 5½
Chicken eggs: 20g
Duck eggs: 35g
There are 15 two yen coins
and 25 five yen coins