Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Omar, an artist, plans to paint and sell some miniature paintings. He just bought some brushes for $\$5$ , and paint and canvas for each painting costs $\$67$ ; he will sell each painting for $\$68$ . Once Omar sells a certain number of his paintings, he will be breaking even. How much will Omar have earned? How many paintings will that be? $\newline$ Omar's expenses and receipts will both total $\$\_\_\_\_$ when he has sold $\_\_\_$ of his paintings.

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Omar, an artist, plans to paint and sell some miniature paintings. He just bought some brushes for

\$5

, and paint and canvas for each painting costs

\$67

; he will sell each painting for

\$68

. Once Omar sells a certain number of his paintings, he will be breaking even. How much will Omar have earned? How many paintings will that be?

\newline

Omar's expenses and receipts will both total

\$\_\_\_\_

when he has sold

\_\_\_

of his paintings.

Define Expenses Equation: Let $x$ represent the number of paintings Omar sells. His expenses include the initial cost of brushes and the cost of paint and canvas for each painting. His income comes from selling each painting. We need to find the point where his expenses equal his income, which is the break-even point.
Calculate Income: The cost of brushes is a one-time expense of $\$5$ . The cost of paint and canvas for each painting is $\$67$ , so the total cost for $x$ paintings is $\$67x$ . Therefore, the equation for Omar's expenses ( $E$ ) is: $\newline$ $E = 67x + 5$
Set Break-Even Point: Omar sells each painting for $\$68$ , so his income $I$ from selling $x$ paintings is: $\newline$ $I = 68x$
Solve for x: To find the break-even point, we set the expenses equal to the income: $67x + 5 = 68x$
Substitute $x$ into Income: Now we solve for $x$ by subtracting $67x$ from both sides of the equation: $\newline$ $67x + 5 - 67x = 68x - 67x$ $\newline$ $5 = x$
Final Answer: We found that $x = 5$ , which means Omar needs to sell $5$ paintings to break even. To find out how much Omar will have earned at this point, we substitute $x$ into the income equation: $\newline$ $I = 68x$ $\newline$ $I = 68(5)$ $\newline$ $I = \$(340)$
Final Answer: We found that $x = 5$ , which means Omar needs to sell $5$ paintings to break even. To find out how much Omar will have earned at this point, we substitute $x$ into the income equation: $\newline$ $I = 68x$ $\newline$ $I = 68(5)$ $\newline$ $I = \$(340)$ Omar's expenses and receipts will both total \$$340$ when he has sold $5$ of his paintings. This answers the question prompt.