Answer:
1. a. b = - 8
b. x = 8
c. x = 11
d. x = 5
2. 12 soccer balls and 8 basketballs can be purchased.
Step by step explanation
a. ![- 14 + 6b + 7 - 2b = 1 + 5b](https://tex.z-dn.net/?f=%20-%2014%20%2B%206b%20%2B%207%20-%202b%20%3D%201%20%2B%205b)
Calculate the sum
![- 7 + 6b - 2b = 1 + 5b](https://tex.z-dn.net/?f=%20%20%20-%207%20%2B%206b%20-%202b%20%3D%201%20%2B%205b)
Collect like terms
![7 + 4b = 1 + 5b](https://tex.z-dn.net/?f=%207%20%2B%204b%20%3D%201%20%2B%205b)
Move variable to L.H.S and change it's sign
Similarly, Move constant to R.H.S and change its sign
![4b - 5b = 1 + 7](https://tex.z-dn.net/?f=4b%20-%205b%20%3D%201%20%2B%207)
Collect like terms
![- b = 8](https://tex.z-dn.net/?f=%20-%20b%20%3D%208)
Change the signs on both sides of the equation
![b = - 8](https://tex.z-dn.net/?f=b%20%3D%20%20-%208)
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b. ![\frac{5x + 10}{ - 6} = - 5](https://tex.z-dn.net/?f=%20%5Cfrac%7B5x%20%20%20%2B%2010%7D%7B%20-%206%7D%20%20%3D%20%20-%205)
Apply cross product property
![5x + 10 = - 5 \times ( - 6)](https://tex.z-dn.net/?f=5x%20%2B%2010%20%3D%20%20-%205%20%5Ctimes%20%20%28%20-%206%29)
Multiply the numbers
![5x + 10 = 30](https://tex.z-dn.net/?f=5x%20%2B%2010%20%3D%2030)
Move constant to R.H.S and change its sign
![5x = 30 - 10](https://tex.z-dn.net/?f=5x%20%3D%2030%20-%2010)
Calculate the difference
![5x = 20](https://tex.z-dn.net/?f=5x%20%3D%2020)
Divide both sides of the equation by 5
![\frac{5x}{5} = \frac{20}{5}](https://tex.z-dn.net/?f=%20%5Cfrac%7B5x%7D%7B5%7D%20%20%20%3D%20%5Cfrac%7B20%7D%7B5%7D%20)
Calculate
![x = 4](https://tex.z-dn.net/?f=x%20%3D%204)
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c. ![- 15 = \frac{ - 8x - 17}{7}](https://tex.z-dn.net/?f=%20-%2015%20%3D%20%20%5Cfrac%7B%20-%208x%20-%2017%7D%7B7%7D%20)
Apply cross product property
![- 15 \times 7 = - 8x - 17](https://tex.z-dn.net/?f=%20-%2015%20%5Ctimes%207%20%3D%20%20-%208x%20-%2017)
Multiply the numbers
![- 105 = - 8x - 17](https://tex.z-dn.net/?f=%20-%20105%20%3D%20%20-%208x%20-%2017)
Swap the sides of the equation
![- 8x - 17 = - 105](https://tex.z-dn.net/?f=%20-%208x%20-%2017%20%3D%20%20-%20105)
Move constant to R.H.S and change its sign
![- 8x = - 105 + 17](https://tex.z-dn.net/?f=%20-%208x%20%3D%20%20-%20105%20%2B%2017)
Calculate
![- 8x = - 88](https://tex.z-dn.net/?f=%20-%208x%20%3D%20%20-%2088)
Change the signs on both sides of the equation
![8x = 88](https://tex.z-dn.net/?f=8x%20%3D%2088)
Divide both sides of the equation by 8
![\frac{8x}{8} = \frac{88}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B8x%7D%7B8%7D%20%20%3D%20%20%5Cfrac%7B88%7D%7B8%7D%20)
Calculate
![x = 11](https://tex.z-dn.net/?f=x%20%3D%2011)
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D. ![5 = 6x + 5(x - 10)](https://tex.z-dn.net/?f=5%20%3D%206x%20%2B%205%28x%20-%2010%29)
Distribute 5 through the parentheses
![5 = 6x + 5x - 50](https://tex.z-dn.net/?f=5%20%3D%206x%20%2B%205x%20-%2050)
Collect like terms
![5 = 11x - 50](https://tex.z-dn.net/?f=5%20%3D%2011x%20-%2050)
Swap both sides of the equation
![11x - 50 = 5](https://tex.z-dn.net/?f=11x%20-%2050%20%3D%205)
Move constant to R.H.S and change its sign
![11x = 5 + 50](https://tex.z-dn.net/?f=11x%20%3D%205%20%2B%2050)
Calculate the sum
![11x = 55](https://tex.z-dn.net/?f=11x%20%3D%2055)
Divide both sides of the equation by 11
![\frac{11x}{11} = \frac{55}{11}](https://tex.z-dn.net/?f=%20%5Cfrac%7B11x%7D%7B11%7D%20%20%3D%20%20%5Cfrac%7B55%7D%7B11%7D%20)
Calculate
![x = 5](https://tex.z-dn.net/?f=x%20%3D%205)
------------------------------------------------------------------
2.
Solution,
No.of students in soccer = x
No.of students in basketball = y
Total no.of students = 20
i.e x + y = 20 → equation ( i )
Cost of soccer ball = $ 7
Cost of basketball = $ 10
Total budget = $ 164
i.e 7x + 10 y = 165 → equation ( ii )
In equation ( i ),
x + y = 20
Move 'y' to R.H.S and change its sign
x = 20 - y
Put the value of x in equation ( i )
![7(20 - y) + 10y = 164](https://tex.z-dn.net/?f=7%2820%20-%20y%29%20%2B%2010y%20%3D%20164)
![140 - 7y + 10y = 164](https://tex.z-dn.net/?f=140%20-%207y%20%2B%2010y%20%3D%20164)
![3y = 164 - 140](https://tex.z-dn.net/?f=3y%20%3D%20164%20-%20140)
![3y = 24](https://tex.z-dn.net/?f=3y%20%3D%2024)
![y = \frac{24}{3}](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B24%7D%7B3%7D%20)
![y = 8](https://tex.z-dn.net/?f=y%20%3D%208)
Now, put the value of y in equation ( i ) ,
x + y = 20
![x + 8 = 20](https://tex.z-dn.net/?f=x%20%2B%208%20%3D%2020)
![x = 20 - 8](https://tex.z-dn.net/?f=x%20%3D%2020%20-%208)
![x = 12](https://tex.z-dn.net/?f=x%20%3D%2012)
Hence, 12 soccer balls and 8 basketballs can be purchased.
Hope this helps...
Best regards!!