Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Beth is creating beaded jewelry to give to her family and friends. For her family, she assembled $2$ bracelets and $5$ necklaces, using a total of $370$ beads. For her friends, she assembled $7$ bracelets and $5$ necklaces, using a total of $595$ beads. Assuming she uses a consistent number of beads for every bracelet and necklace, how many beads is she using for each? $\newline$ Beth uses $\_$ beads for each bracelet and $\_$ beads for each necklace.

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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

Beth is creating beaded jewelry to give to her family and friends. For her family, she assembled

2

bracelets and

5

necklaces, using a total of

370

beads. For her friends, she assembled

7

bracelets and

5

necklaces, using a total of

595

beads. Assuming she uses a consistent number of beads for every bracelet and necklace, how many beads is she using for each?

\newline

Beth uses

\_

beads for each bracelet and

\_

beads for each necklace.

Equations setup: Let's denote the number of beads used for each bracelet as $b$ and the number of beads used for each necklace as $n$ . For her family, Beth assembled $2$ bracelets and $5$ necklaces, using a total of $370$ beads. This gives us the equation $2b + 5n = 370$ .
Variable elimination: For her friends, she assembled $7$ bracelets and $5$ necklaces, using a total of $595$ beads. This gives us the equation $7b + 5n = 595$ .
Solving for $b$ : We now have a system of two equations. We need to eliminate one of the variables, $b$ or $n$ . We choose to eliminate $n$ because its coefficients are the same in both equations.
Substitute $b$ into first equation: To eliminate $n$ , we subtract the first equation from the second equation. This gives us $7b + 5n - (2b + 5n) = 595 - 370$ . Simplifying, we get $5b = 225$ .
Solving for n: We divide both sides of the equation by $5$ to solve for $b$ . This gives us $b = \frac{225}{5}$ , which simplifies to $b = 45$ .
Solving for n: We divide both sides of the equation by $5$ to solve for $b$ . This gives us $b = \frac{225}{5}$ , which simplifies to $b = 45$ .We substitute $b = 45$ into the first equation and solve for $n$ . This gives us $2(45) + 5n = 370$ . Simplifying, we get $90 + 5n = 370$ .
Solving for n: We divide both sides of the equation by $5$ to solve for $b$ . This gives us $b = \frac{225}{5}$ , which simplifies to $b = 45$ . We substitute $b = 45$ into the first equation and solve for $n$ . This gives us $2(45) + 5n = 370$ . Simplifying, we get $90 + 5n = 370$ . We subtract $90$ from both sides of the equation to solve for $n$ . This gives us $b$ $0$ , which simplifies to $b$ $1$ .
Solving for n: We divide both sides of the equation by $5$ to solve for $b$ . This gives us b = rac{225}{5}, which simplifies to $b = 45$ . We substitute $b = 45$ into the first equation and solve for $n$ . This gives us $2(45) + 5n = 370$ . Simplifying, we get $90 + 5n = 370$ . We subtract $90$ from both sides of the equation to solve for $n$ . This gives us $b$ $0$ , which simplifies to $b$ $1$ . We divide both sides of the equation by $5$ to solve for $n$ . This gives us $b$ $4$ , which simplifies to $b$ $5$ .