Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ For a project in statistics class, a pair of students decided to invest in two companies, one that produces software and one that does biotechnology research. Jill purchased $82$ shares in the software company and $82$ shares in the biotech firm, which cost a total of $\$5,822$ . At the same time, Kate invested a total of $\$5,978$ in $95$ shares in the software company and $82$ shares in the biotech firm. How much did each share cost? $\newline$ Each share in the software company cost $\$\_$ , and each share in the biotech firm cost $\$\_$ .

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Solve a system of equations using elimination: word problems

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

\newline

For a project in statistics class, a pair of students decided to invest in two companies, one that produces software and one that does biotechnology research. Jill purchased

82

shares in the software company and

82

shares in the biotech firm, which cost a total of

\$5,822

. At the same time, Kate invested a total of

\$5,978

95

shares in the software company and

82

shares in the biotech firm. How much did each share cost?

\newline

Each share in the software company cost

\$\_

, and each share in the biotech firm cost

\$\_

Equations setup: Let's denote the cost of one share in the software company as $S$ dollars and the cost of one share in the biotech firm as $B$ dollars. We can write two equations based on the information given: $\newline$ $1$ . For Jill's investment: $82S + 82B = 5822$ $\newline$ $2$ . For Kate's investment: $95S + 82B = 5978$
Elimination method: To solve this system using elimination, we need to eliminate one of the variables. We can do this by multiplying the first equation by $-1$ to get the coefficients of $B$ to be opposites: $\newline$ $-1 \times (82S + 82B) = -1 \times 5822$ $\newline$ This gives us: $-82S - 82B = -5822$
Solving for S: Now we add this new equation to the second equation to eliminate B: $\newline$ $(-82S - 82B) + (95S + 82B) = (-5822) + 5978$ $\newline$ This simplifies to: $13S = 156$
Substitute $S$ into equation: We divide both sides of the equation by $13$ to solve for $S$ : $\frac{13S}{13} = \frac{156}{13}$ $S = 12$
Calculate value of B: Now that we have the value of S, we can substitute it back into one of the original equations to solve for B. We'll use the first equation: $\newline$ $82S + 82B = 5822$ $\newline$ $82(12) + 82B = 5822$
Final solution for B: We calculate the value of $82 \times 12$ and subtract it from $5822$ to solve for B: $\newline$ $984 + 82B = 5822$ $\newline$ $82B = 5822 - 984$ $\newline$ $82B = 4838$
Final solution for B: We calculate the value of $82 \times 12$ and subtract it from $5822$ to solve for B: $\newline$ $984 + 82B = 5822$ $\newline$ $82B = 5822 - 984$ $\newline$ $82B = 4838$ Finally, we divide both sides of the equation by $82$ to find the value of B: $\newline$ $\frac{82B}{82} = \frac{4838}{82}$ $\newline$ $B = 59$