which expression is equivalent to $\frac{8x(x - 7)-5(x - 7)}{2x - 14}$, where $x7$? a) $\frac{x - 7}{5}$ b) $\frac{8x-5}{2}$ c) $\frac{8x^{2}-5x - 14}{2x - 14}$ d) $\frac{8x^{2}-5x - 77}{2x - 14}$ keenan made 32 cups of vegetable broth. keenan then filled $x$ small jars and $y$ large jars with all the vegetable broth he made. the equation $3x + 5y=32$ represents this situation. which is the best interpretation of $5y$ in this context? a) the number of large jars keenan filled b) the number of small jars keenan filled c) the total number of cups of vegetable broth in the large jars d) the total number of cups of vegetable broth in the small jars the function $f$ is defined by $f(x)=(-8)(2)^{x}+22$. what is the $y$-intercept of the graph of $y = f(x)$ in the $xy$-plane? a) $(0,14)$ b) $(0,2)$ c) $(0,22)$ d) $(0,-8)$ a circle in the $xy$-plane has a diameter with endpoints $(2,4)$ and $(2,14)$. an equation of this circle is $(x - 2)^{2}+(y - 9)^{2}=r^{2}$, where $r$ is a positive constant. what is the value of $r$? line $l$ is defined by $3y+12x = 5$. line $m$ is perpendicular to line $l$ in the $xy$-plane. what is the slope of line $m$?

Question

which expression is equivalent to $\frac{8x(x - 7)-5(x - 7)}{2x - 14}$, where $x>7$? a) $\frac{x - 7}{5}$ b) $\frac{8x-5}{2}$ c) $\frac{8x^{2}-5x - 14}{2x - 14}$ d) $\frac{8x^{2}-5x - 77}{2x - 14}$ keenan made 32 cups of vegetable broth. keenan then filled $x$ small jars and $y$ large jars with all the vegetable broth he made. the equation $3x + 5y=32$ represents this situation. which is the best interpretation of $5y$ in this context? a) the number of large jars keenan filled b) the number of small jars keenan filled c) the total number of cups of vegetable broth in the large jars d) the total number of cups of vegetable broth in the small jars the function $f$ is defined by $f(x)=(-8)(2)^{x}+22$. what is the $y$-intercept of the graph of $y = f(x)$ in the $xy$-plane? a) $(0,14)$ b) $(0,2)$ c) $(0,22)$ d) $(0,-8)$ a circle in the $xy$-plane has a diameter with endpoints $(2,4)$ and $(2,14)$. an equation of this circle is $(x - 2)^{2}+(y - 9)^{2}=r^{2}$, where $r$ is a positive constant. what is the value of $r$? line $l$ is defined by $3y+12x = 5$. line $m$ is perpendicular to line $l$ in the $xy$-plane. what is the slope of line $m$?

Accepted Answer

### First Question
# Explanation:
## Step1: Simplify the numerator
First, expand $8x(x - 7)-8(x - 7)$:
$$
\begin{align*}
8x(x - 7)-8(x - 7)&=(x - 7)(8x-8)\
&=8(x - 7)(x - 1)
\end{align*}
$$
The denominator is $2x-14 = 2(x - 7)$.
## Step2: Cancel out common factors
Since $x>7$, we can cancel out the non - zero factor $(x - 7)$ from the numerator and the denominator.
$\frac{8(x - 7)(x - 1)}{2(x - 7)}=\frac{8(x - 1)}{2}=4(x - 1)=\frac{8x-8}{2}$
# Answer: B

### Second Question
# Brief Explanations:
In the equation $3x + 5y=32$, where $x$ is the number of small jars and $y$ is the number of large jars, and the coefficients represent the number of cups of vegetable broth each jar size holds. Since $5$ is the number of cups of vegetable broth in each large jar and $y$ is the number of large jars, $5y$ represents the total number of cups of vegetable broth in the large jars.
# Answer: C

### Third Question
# Explanation:
## Step1: Find the y - intercept
The y - intercept of the graph of $y = f(x)$ is found by setting $x = 0$.
Given $f(x)=(-8)(2)^{x}+22$, when $x = 0$, we know that $2^{0}=1$.
So $f(0)=(-8)\times1 + 22=-8 + 22=14$.
The y - intercept is the point $(0,14)$.
# Answer: A

### Fourth Question
# Explanation:
## Step1: Find the center of the circle
The center of a circle with diameter endpoints $(x_1,y_1)=(2,4)$ and $(x_2,y_2)=(2,14)$ has x - coordinate $x=\frac{x_1 + x_2}{2}=\frac{2+2}{2}=2$ and y - coordinate $y=\frac{y_1 + y_2}{2}=\frac{4 + 14}{2}=9$.
The equation of the circle is $(x - 2)^{2}+(y - 9)^{2}=r^{2}$.
## Step2: Calculate the radius
The radius $r$ is the distance from the center $(2,9)$ to either endpoint. Using the distance formula $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$ with the center $(2,9)$ and the point $(2,4)$ (we could also use the other endpoint), $x_1 = 2,y_1 = 9,x_2 = 2,y_2 = 4$.
$r=\sqrt{(2 - 2)^{2}+(4 - 9)^{2}}=\sqrt{0+(-5)^{2}} = 5$
# Answer: 5

### Fifth Question
# Explanation:
## Step1: Rewrite line $\ell$ in slope - intercept form
Given the equation of line $\ell$: $3y+12x = 5$, rewrite it as $y=-4x+\frac{5}{3}$. The slope of line $\ell$, denoted as $m_1=-4$.
## Step2: Find the slope of the perpendicular line
If two lines are perpendicular, the product of their slopes $m_1\times m_2=-1$. Let the slope of line $n$ be $m_2$.
Since $m_1=-4$, then $-4m_2=-1$, so $m_2=\frac{1}{4}$
# Answer: $\frac{1}{4}$

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QUESTION IMAGE

Question

Explanation:

First Question

Step1: Simplify the numerator

Step2: Cancel out common factors

Step1: Find the y - intercept

Answer:

Second Question