D) Later, Jerry purchased items for his garden in his new yard. He purchased plants for $15 and bags of soil for $15. He used the expression: 15(x + y)
His friend Valerie says that she would use the expression: 15x + 15y
Are the expressions equivalent? Justify your reasoning using number properties.
first off, let's split the triplet into two equations, then from there on we'll do substitution.
![\cfrac{y}{x-z}=\cfrac{x}{y}=\cfrac{x+y}{z}\implies \begin{cases} \cfrac{y}{x-z}=\cfrac{x}{y}\\[2em] \cfrac{x}{y}=\cfrac{x+y}{z} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 1st equation}}{\cfrac{y}{x-z}=\cfrac{x}{y}\implies }y^2=\underline{x^2-xz} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 2nd equation}}{\cfrac{x}{y}=\cfrac{x+y}{z}\implies }xz=xy+y^2\implies \stackrel{\textit{substituting for }y^2}{xz=xy+(\underline{x^2-xz})}](https://tex.z-dn.net/?f=%5Ccfrac%7By%7D%7Bx-z%7D%3D%5Ccfrac%7Bx%7D%7By%7D%3D%5Ccfrac%7Bx%2By%7D%7Bz%7D%5Cimplies%20%5Cbegin%7Bcases%7D%20%5Ccfrac%7By%7D%7Bx-z%7D%3D%5Ccfrac%7Bx%7D%7By%7D%5C%5C%5B2em%5D%20%5Ccfrac%7Bx%7D%7By%7D%3D%5Ccfrac%7Bx%2By%7D%7Bz%7D%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Busing%20the%201st%20equation%7D%7D%7B%5Ccfrac%7By%7D%7Bx-z%7D%3D%5Ccfrac%7Bx%7D%7By%7D%5Cimplies%20%7Dy%5E2%3D%5Cunderline%7Bx%5E2-xz%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Busing%20the%202nd%20equation%7D%7D%7B%5Ccfrac%7Bx%7D%7By%7D%3D%5Ccfrac%7Bx%2By%7D%7Bz%7D%5Cimplies%20%7Dxz%3Dxy%2By%5E2%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bsubstituting%20for%20%7Dy%5E2%7D%7Bxz%3Dxy%2B%28%5Cunderline%7Bx%5E2-xz%7D%29%7D)
![2xz=xy+x^2\implies 2xz=x(y+x)\implies \cfrac{2xz}{x}=y+x \\\\\\ 2z=y+x\implies 2=\cfrac{y+x}{z}\implies 2=\cfrac{x+y}{z} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{}{ \begin{cases} \cfrac{y}{x-z}=\cfrac{x}{y}\\[2em] \cfrac{x}{y}=\cfrac{x+y}{z} \end{cases}}\implies \begin{cases} \cfrac{y}{x-z}=\cfrac{x}{y}\\[2em] \cfrac{x}{y}=2 \end{cases}\implies \begin{cases} \cfrac{y}{x-z}=2\\[2em] \cfrac{x}{y}=2 \end{cases}](https://tex.z-dn.net/?f=2xz%3Dxy%2Bx%5E2%5Cimplies%202xz%3Dx%28y%2Bx%29%5Cimplies%20%5Ccfrac%7B2xz%7D%7Bx%7D%3Dy%2Bx%20%5C%5C%5C%5C%5C%5C%202z%3Dy%2Bx%5Cimplies%202%3D%5Ccfrac%7By%2Bx%7D%7Bz%7D%5Cimplies%202%3D%5Ccfrac%7Bx%2By%7D%7Bz%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cstackrel%7B%7D%7B%20%5Cbegin%7Bcases%7D%20%5Ccfrac%7By%7D%7Bx-z%7D%3D%5Ccfrac%7Bx%7D%7By%7D%5C%5C%5B2em%5D%20%5Ccfrac%7Bx%7D%7By%7D%3D%5Ccfrac%7Bx%2By%7D%7Bz%7D%20%5Cend%7Bcases%7D%7D%5Cimplies%20%5Cbegin%7Bcases%7D%20%5Ccfrac%7By%7D%7Bx-z%7D%3D%5Ccfrac%7Bx%7D%7By%7D%5C%5C%5B2em%5D%20%5Ccfrac%7Bx%7D%7By%7D%3D2%20%5Cend%7Bcases%7D%5Cimplies%20%5Cbegin%7Bcases%7D%20%5Ccfrac%7By%7D%7Bx-z%7D%3D2%5C%5C%5B2em%5D%20%5Ccfrac%7Bx%7D%7By%7D%3D2%20%5Cend%7Bcases%7D)
that of course, is only true if x + y, or our numerator doesn't turn into 0, if it does then our fraction becomes 0 and our equation goes south. Keeping in mind that x,y and z are numeric values that correlate like so.
Answer:
0
Step-by-step explanation:
It's a horizontal line, it has no slope.
Answer:
<h3>explicit formula- a(n)=6+4(n-1)</h3><h3>recursive formula- a(n)=a(n)-1 +4; a(1)=6</h3>
Step-by-step explanation:
