Let's assume
number of small notebooks =x
number of large notebooks =y
we are given
total number of notebooks =31
so, we get
![x+y=31](https://tex.z-dn.net/?f=%20x%2By%3D31%20)
now, we can solve for y
![y=31-x](https://tex.z-dn.net/?f=%20y%3D31-x%20)
we have
Small notebooks cost $3.50 and large notebooks cost $5.00
she has $134 to spend
so, we get
![3.50x+5y=134](https://tex.z-dn.net/?f=%203.50x%2B5y%3D134%20)
now, we can plug y
![3.50x+5(31-x)=134](https://tex.z-dn.net/?f=%203.50x%2B5%2831-x%29%3D134%20)
![3.50x+5*31-5x=134](https://tex.z-dn.net/?f=%203.50x%2B5%2A31-5x%3D134%20)
![-1.5x+155=134](https://tex.z-dn.net/?f=%20-1.5x%2B155%3D134%20)
![-1.5x+155-155=134-155](https://tex.z-dn.net/?f=%20-1.5x%2B155-155%3D134-155%20)
![x=14](https://tex.z-dn.net/?f=%20x%3D14%20)
now, we can find y
![y=31-14](https://tex.z-dn.net/?f=%20y%3D31-14%20)
![y=17](https://tex.z-dn.net/?f=%20y%3D17%20)
so,
number of small notebooks is 14
number of large notebooks is 17.............Answer
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Answer:
![x = \frac{ 5 \ + \ \sqrt{31}}{2} \ , \ x = \frac{ 5 \ - \ \sqrt{31}}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B%205%20%5C%20%2B%20%5C%20%5Csqrt%7B31%7D%7D%7B2%7D%20%5C%20%2C%20%5C%20x%20%3D%20%5Cfrac%7B%205%20%5C%20-%20%5C%20%5Csqrt%7B31%7D%7D%7B2%7D)
Step-by-step explanation:
![2x^2 - 10x - 3 = 0 \\\\a = 2 \ , b = - 10 \ , \ c = - 3 \\\\x = \frac{-b^2\ \pm \ \sqrt{b^2 - 4ac}}{2a}\\\\x = \frac{10 \ \pm \sqrt{(-10)^2 - ( 4 \times 2 \times -3)} }{2 \times 2}\\\\x = \frac{10 \ \pm \sqrt{(100 - ( -24 )} }{4}\\\\x = \frac{10 \ \pm \sqrt{(100 + 24 } }{4}\\\\x = \frac{ 10 \ \pm \sqrt{124}}{4}\\\\x = \frac{ 10 \ \pm \sqrt{4 \times 31}}{4}\\\\x = \frac{ 10 \ \pm \sqrt{2^2 \times 31}}{4}\\\\x = \frac{ 10 \ \pm2 \sqrt{31}}{4}\\\\x = \frac{ 5 \ \pm\sqrt{31}}{2}\\\\](https://tex.z-dn.net/?f=2x%5E2%20-%2010x%20-%203%20%3D%200%20%5C%5C%5C%5Ca%20%3D%202%20%5C%20%2C%20b%20%3D%20-%2010%20%5C%20%2C%20%5C%20c%20%3D%20%20-%203%20%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B-b%5E2%5C%20%20%5Cpm%20%5C%20%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B2a%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B10%20%5C%20%5Cpm%20%5Csqrt%7B%28-10%29%5E2%20-%20%28%204%20%5Ctimes%202%20%5Ctimes%20-3%29%7D%20%7D%7B2%20%5Ctimes%202%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B10%20%5C%20%5Cpm%20%5Csqrt%7B%28100%20-%20%28%20-24%20%29%7D%20%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B10%20%5C%20%5Cpm%20%5Csqrt%7B%28100%20%2B%2024%20%7D%20%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B%2010%20%5C%20%5Cpm%20%5Csqrt%7B124%7D%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B%2010%20%5C%20%5Cpm%20%5Csqrt%7B4%20%5Ctimes%2031%7D%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B%2010%20%5C%20%5Cpm%20%5Csqrt%7B2%5E2%20%5Ctimes%2031%7D%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B%2010%20%5C%20%5Cpm2%20%5Csqrt%7B31%7D%7D%7B4%7D%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B%205%20%5C%20%5Cpm%5Csqrt%7B31%7D%7D%7B2%7D%5C%5C%5C%5C)
![x = \frac{ 5 \ + \ \sqrt{31}}{2} \ , \ x = \frac{ 5 \ - \ \sqrt{31}}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B%205%20%5C%20%2B%20%5C%20%5Csqrt%7B31%7D%7D%7B2%7D%20%5C%20%2C%20%5C%20x%20%3D%20%5Cfrac%7B%205%20%5C%20-%20%5C%20%5Csqrt%7B31%7D%7D%7B2%7D)
4(x+5y)(x-3y)
I don't know what you mean by placing the factors on the grid