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Solve the pair of equations with

{:[{[(1)/(y)+(1)/(2x)=1],[(2)/(5y)+(4)/(5x)=1]:}],[x=◻],[y=◻]:}

Solve the pair of equations with $\newline$ $\begin{array}{l} \left\{\begin{array}{l} \frac{1}{y}+\frac{1}{2 x}=1 \\ \frac{2}{5 y}+\frac{4}{5 x}=1 \end{array}\right. \\ x=\square \\ y=\square \end{array}$

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Compare linear and exponential growth

Full solution

Q. Solve the pair of equations with

\newline

\begin{array}{l} \left\{\begin{array}{l} \frac{1}{y}+\frac{1}{2 x}=1 \\ \frac{2}{5 y}+\frac{4}{5 x}=1 \end{array}\right. \\ x=\square \\ y=\square \end{array}

Write Equations: Write down the equations. $\newline$ Equation $1$ : $\frac{1}{y} + \frac{1}{2x} = 1$ $\newline$ Equation $2$ : $\frac{2}{5y} + \frac{4}{5x} = 1$
Multiply Equation $1$ : Multiply Equation $1$ by $2x$ to clear the fraction. $\newline$ $2x \cdot \frac{1}{y} + 2x \cdot \frac{1}{2x} = 2x \cdot 1$ $\newline$ $\frac{2x}{y} + 1 = 2x$
Rearrange Equation $1$ : Rearrange Equation $1$ to solve for $\frac{2x}{y}$ . $\newline$ $\frac{2x}{y} = 2x - 1$
Multiply Equation $2$ : Multiply Equation $2$ by $5xy$ to clear the fraction. $\newline$ $5xy \cdot \frac{2}{5y} + 5xy \cdot \frac{4}{5x} = 5xy \cdot 1$ $\newline$ $2x + 4y = 5xy$
Substitute in Equation $2$ : Substitute $\frac{2x}{y} = 2x - 1$ into Equation $2$ . $\newline$ $2x + 4y = 5xy$
Solve for y: Solve for $y$ in terms of $x$ . $\newline$ $4y = 5xy - 2x$ $\newline$ $4y = x(5y - 2)$
Isolate y: Isolate $y$ . $\newline$ $4y = 5xy - 2x$ $\newline$ $4y = x(5y - 2)$ $\newline$ Divide both sides by $5y - 2$ : $\newline$ $y = \frac{2x}{5x - 4}$
Substitute for y: Substitute $y = \frac{2x}{5x - 4}$ back into Equation $1$ . $\newline$ $\frac{1}{\frac{2x}{5x - 4}} + \frac{1}{2x} = 1$ $\newline$ $\frac{5x - 4}{2x} + \frac{1}{2x} = 1$ $\newline$ $\frac{5x - 4 + 1}{2x} = 1$ $\newline$ $\frac{5x - 3}{2x} = 1$
Solve for x: Solve for $x$ . $\newline$ $5x - 3 = 2x$ $\newline$ $3x = 3$ $\newline$ $x = 1$
Substitute for y: Substitute $x = 1$ back into $y = \frac{2x}{5x - 4}$ . $\newline$ $y = \frac{2 \cdot 1}{5 \cdot 1 - 4}$ $\newline$ $y = \frac{2}{1}$ $\newline$ $y = 2$

More problems from Compare linear and exponential growth

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Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

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Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

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g(x) = \underline{\hspace{3em}}

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f(t)=-2t-7

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\newline

\left[\text{f(t) decreases by } 2\right]

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\left[\text{f(t) increases by } 2\right]

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\left[\text{f(t) increases by } 200\%\right]

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\left[\text{f(t) decreases by a factor of } 2\right]

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g(t)=9^t

change over the interval from

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\left[\text{g(t) decreases by a factor of } 81\right]

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\left[\text{g(t) increases by a factor of } 81\right]

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Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 6 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 6 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 6 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

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r'(3)=

Posted 7 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 7 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 7 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 7 months ago

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