Answer:
1. The equations is inconsistent and has no solution
2. The equations is consistent and has no solution
3. A pen costs $1.50 and a notebook costs $2.00
Step-by-step explanation:
Solving
and ![3y = 9 - 7x](https://tex.z-dn.net/?f=3y%20%3D%209%20-%207x)
--- Equation 1
--- Equation 2
Add 7x to both sides in equation 2
![7x + 3y = 9 - 7x + 7x](https://tex.z-dn.net/?f=7x%20%2B%203y%20%3D%209%20-%207x%20%2B%207x)
--- Equation 3
Subtract equation 3 from 1
![(7x + 3y = 10) - (7x + 3y = 9)](https://tex.z-dn.net/?f=%287x%20%2B%203y%20%3D%2010%29%20-%20%287x%20%2B%203y%20%3D%209%29)
![7x - 7x + 3y - 3y = 10 - 9](https://tex.z-dn.net/?f=7x%20-%207x%20%2B%203y%20-%203y%20%3D%2010%20-%209)
![0 + 0 = 1](https://tex.z-dn.net/?f=0%20%2B%200%20%3D%201)
![0 \neq 1](https://tex.z-dn.net/?f=0%20%5Cneq%201)
The equations is inconsistent and has no solution
Solving
and ![12x + 4y = 16](https://tex.z-dn.net/?f=12x%20%2B%204y%20%3D%2016)
-- Equation 1
---- Equation 2
Make y the subject of formula in equation 1
![y = 4 - 3x](https://tex.z-dn.net/?f=y%20%3D%204%20-%203x)
Substitute 4 - 3x for y in equation 2
![12x + 4(4 - 3x) = 16](https://tex.z-dn.net/?f=12x%20%2B%204%284%20-%203x%29%20%3D%2016)
![12x + 16 - 12x = 16](https://tex.z-dn.net/?f=12x%20%2B%2016%20-%2012x%20%3D%2016)
Collect Like Terms
![12x - 12x = 16 - 16](https://tex.z-dn.net/?f=12x%20-%2012x%20%3D%2016%20-%2016)
![0 = 0](https://tex.z-dn.net/?f=0%20%3D%200)
The equations is consistent and has no solution
3. Solving <em>John and Harry</em>
Given;
<em>Represent Pen with P and Notes with N</em>
John: ![3P + 8N = 20.5](https://tex.z-dn.net/?f=3P%20%2B%208N%20%3D%2020.5)
Harry: ![4P + 5N = 16.00](https://tex.z-dn.net/?f=4P%20%2B%205N%20%3D%2016.00)
Required
Find P and N
Make P the subject of formula in equation 2
![4P = 16 - 5N](https://tex.z-dn.net/?f=4P%20%3D%2016%20-%205N)
![P = \frac{16 - 5N}{4}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B16%20-%205N%7D%7B4%7D)
Substitute
in equation 1
![3P + 8N = 20.5](https://tex.z-dn.net/?f=3P%20%2B%208N%20%3D%2020.5)
![3(\frac{16 - 5N}{4}) + 8N = 20.5](https://tex.z-dn.net/?f=3%28%5Cfrac%7B16%20-%205N%7D%7B4%7D%29%20%2B%208N%20%3D%2020.5)
Open the bracket
![\frac{48 - 15N}{4} + 8N = 20.5](https://tex.z-dn.net/?f=%5Cfrac%7B48%20-%2015N%7D%7B4%7D%20%2B%208N%20%3D%2020.5)
Take LCM
![\frac{48 - 15N + 32N}{4} = 20.5](https://tex.z-dn.net/?f=%5Cfrac%7B48%20-%2015N%20%2B%2032N%7D%7B4%7D%20%3D%2020.5)
![\frac{48 + 17N}{4} = 20.5](https://tex.z-dn.net/?f=%5Cfrac%7B48%20%2B%2017N%7D%7B4%7D%20%3D%2020.5)
Multiply both sides by 4
![4 * \frac{48 + 17N}{4} = 20.5 * 4](https://tex.z-dn.net/?f=4%20%2A%20%5Cfrac%7B48%20%2B%2017N%7D%7B4%7D%20%3D%2020.5%20%2A%204)
![48 + 17N = 82](https://tex.z-dn.net/?f=48%20%2B%2017N%20%3D%2082)
Subtract 48 from both sides
![48 - 48 + 17N = 82 - 48](https://tex.z-dn.net/?f=48%20-%2048%20%2B%2017N%20%3D%2082%20-%2048)
![17N = 82 - 48](https://tex.z-dn.net/?f=17N%20%3D%2082%20-%2048)
![17N = 34](https://tex.z-dn.net/?f=17N%20%3D%2034)
Divide both sides by 17
![17N/17 = 34/17](https://tex.z-dn.net/?f=17N%2F17%20%3D%2034%2F17)
![N = 34/17](https://tex.z-dn.net/?f=N%20%3D%2034%2F17)
![N = 2](https://tex.z-dn.net/?f=N%20%3D%202)
Substitute 2 for N in ![P = \frac{16 - 5N}{4}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B16%20-%205N%7D%7B4%7D)
![P = \frac{16 - 5 * 2}{4}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B16%20-%205%20%2A%202%7D%7B4%7D)
![P = \frac{16 - 10}{4}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B16%20-%2010%7D%7B4%7D)
![P = \frac{6}{4}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B6%7D%7B4%7D)
![P = 1.5](https://tex.z-dn.net/?f=P%20%3D%201.5)
<em>Hence, a pen costs $1.50 and a notebook costs $2.00</em>