Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Bernie is a shoe salesman, and he works on commission. This week, there is a special incentive to sell shoes and boots by a certain company. Yesterday, Bernie sold $6$ pairs of shoes and $8$ pairs of boots, earning $\$194$ in commission. Today, he sold $4$ pairs of shoes and $1$ pair of boots, earning a total commission of $\$47$ . How much does Bernie earn for the sale of each type of footwear? $\newline$ Bernie earns $\$$ _ for each pair of shoes and $\$$ _ for each pair of boots.

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Solve a system of equations using any method: word problems

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

\newline

Bernie is a shoe salesman, and he works on commission. This week, there is a special incentive to sell shoes and boots by a certain company. Yesterday, Bernie sold

6

pairs of shoes and

8

pairs of boots, earning

\$194

in commission. Today, he sold

4

pairs of shoes and

1

pair of boots, earning a total commission of

\$47

. How much does Bernie earn for the sale of each type of footwear?

\newline

Bernie earns

\$

_____ for each pair of shoes and

\$

_____ for each pair of boots.

Define Earnings: Let's denote the amount Bernie earns for each pair of shoes as $s$ and for each pair of boots as $b$ .
Commission Calculation: According to the problem, Bernie sold $6$ pairs of shoes and $8$ pairs of boots yesterday, earning $\$194$ in commission. This gives us the equation: $\newline$ $6s + 8b = 194$
System of Equations: Today, Bernie sold $4$ pairs of shoes and $1$ pair of boots, earning a total commission of $\$47$ . This gives us the second equation: $\newline$ $4s + b = 47$
Elimination Method: We now have a system of equations to solve: $\newline$ $6s + 8b = 194$ $\newline$ $4s + b = 47$
Substitute and Solve: To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the second equation by $8$ to match the coefficient of $b$ in the first equation: $\newline$ $8$ ( $4$ s + b) = $8$ ( $47$ ) $\newline$ $32$ s + $8$ b = $376$
Substitute and Solve: To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the second equation by $8$ to match the coefficient of $b$ in the first equation: $\newline$ $8$ ( $4$ s + b) = $8$ ( $47$ ) $\newline$ $32$ s + $8$ b = $376$ Now we subtract the first equation from the modified second equation to eliminate $b$ : $\newline$ ( $32$ s + $8$ b) - ( $6$ s + $8$ b) = $376$ - $194$ $\newline$ $32$ s - $6$ s = $376$ - $194$ $\newline$ $26$ s = $182$
Substitute and Solve: To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the second equation by $8$ to match the coefficient of $b$ in the first equation: $\newline$ $8$ ( $4$ s + b) = $8$ ( $47$ ) $\newline$ $32$ s + $8$ b = $376$ Now we subtract the first equation from the modified second equation to eliminate $b$ : $\newline$ ( $32$ s + $8$ b) - ( $6$ s + $8$ b) = $376$ - $194$ $\newline$ $32$ s - $6$ s = $376$ - $194$ $\newline$ $26$ s = $182$ Divide both sides by $26$ to solve for $s$ : $\newline$ $26$ s / $26$ = $182$ / $26$ $\newline$ s = $7$
Substitute and Solve: To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the second equation by $8$ to match the coefficient of $b$ in the first equation: $\newline$ $8$ ( $4$ s + b) = $8$ ( $47$ ) $\newline$ $32$ s + $8$ b = $376$ Now we subtract the first equation from the modified second equation to eliminate $b$ : $\newline$ ( $32$ s + $8$ b) - ( $6$ s + $8$ b) = $376$ - $194$ $\newline$ $32$ s - $6$ s = $376$ - $194$ $\newline$ $26$ s = $182$ Divide both sides by $26$ to solve for $s$ : $\newline$ $26$ s / $26$ = $182$ / $26$ $\newline$ s = $7$ Now that we have the value for $s$ , we can substitute it back into one of the original equations to solve for $b$ . Let's use the second equation: $\newline$ $4$ s + b = $47$ $\newline$ $4$ ( $7$ ) + b = $47$ $\newline$ $28$ + b = $47$
Substitute and Solve: To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the second equation by $8$ to match the coefficient of $b$ in the first equation: $\newline$ $8$ ( $4$ s + b) = $8$ ( $47$ ) $\newline$ $32$ s + $8$ b = $376$ Now we subtract the first equation from the modified second equation to eliminate $b$ : $\newline$ ( $32$ s + $8$ b) - ( $6$ s + $8$ b) = $376$ - $194$ $\newline$ $32$ s - $6$ s = $376$ - $194$ $\newline$ $26$ s = $182$ Divide both sides by $26$ to solve for $s$ : $\newline$ $26$ s / $26$ = $182$ / $26$ $\newline$ s = $7$ Now that we have the value for $s$ , we can substitute it back into one of the original equations to solve for $b$ . Let's use the second equation: $\newline$ $4$ s + b = $47$ $\newline$ $4$ ( $7$ ) + b = $47$ $\newline$ $28$ + b = $47$ Subtract $28$ from both sides to solve for $b$ : $\newline$ b = $47$ - $28$ $\newline$ b = $19$