Question

Leah would like to earn at least $120 per month. She babysits for $5 per hour and
works at an ice cream shop for $8 per hour. Leah cannot work more than a total of
20 hours per month. Let x represent the number of hours Leah babysits and let y
represent the number of hours Leah works at the ice cream shop.

Answer 1

The system of inequalities that represents the given situation:

$x+y\leq 20\\\\5x+8y\geq 120$

Solution:

Given that,

Let x represent the number of hours Leah babysits

Let y  represent the number of hours Leah works at the ice cream shop

Given that,

Leah cannot work more than a total of  20 hours per month

Therefore,

$x+y\leq 20--------- eqn\ 1$

Here we used "less than or equal to" symbol, since she cannot work more than 20 hours

Babysits = $5 per hour

Ice cream shop = $8 per hour

Leah would like to earn at least $120 per month

Thus a inequality is framed as:

$x \times 5 + y \times 8\geq 120\\\\5x+8y\geq 120 ------- eqn\ 2$

Here, we used "greater than or equal to" symbol, since she wants to earn at least $ 120

Thus eqn 1 and eqn 2 are the system of inequalities that represents the given situation

Answer 2

Linear equation in two variables as defined by mathematics refers to an equation having two variables and is basically in the form:

$\rm ax + by=c$

The linear equations for the given questions are:

$\rm x +y=20\\ \\ 5x+8y=120$

Explanation:

Given:

$\begin{aligned} \rm Desired \:Earnings &= \ 120/ month\\ \\ \rm Rate\:of\:babysitting&= \ 5/hour\\ \\ \rm Rate\:of \:working \:in \:shop&= \ 8/hour\\ \\\ \rm Maximum\:hours \:can\:be \:worked&=20\: hours/month\end$

Let hours of babysitting be x and hours of working in ice-cream shop be y. The total working hours can be represented as :

$\begin{aligned} \rm Total\:hours &= Hours\:of \:babysitting + Hours\:of\:working\:in\:shop\\ \\ 20&=x+y\end$

Since total earnings are a product of hours worked and the hourly rate, the earnings can be represented as:

$\begin{aligned} \rm Earnings&= (Rate\times hours\:of\:babysitting)+(Rate \times hours \:of \:working\:in\:the \:shop)\\ \\ \rm 120&=(5\times x)+(8\times y)\\ \\ \rm 120&=5x+8y\end$

Hence the equations are:

$\rm x +y=20\\\\5x+8y=120$

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