Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Valentina is creating beaded jewelry to give to her family and friends. For her family, she assembled $6$ bracelets and $4$ necklaces, using a total of $532$ beads. For her friends, she assembled $4$ bracelets and $7$ necklaces, using a total of $697$ beads. Assuming she uses a consistent number of beads for every bracelet and necklace, how many beads is she using for each? $\newline$ Valentina uses _ beads for each bracelet and _ beads for each necklace.

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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

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

6

bracelets and

4

necklaces, using a total of

532

beads. For her friends, she assembled

4

bracelets and

7

necklaces, using a total of

697

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

\newline

Valentina uses _____ beads for each bracelet and _____ beads for each necklace.

Define Variables: Let's define the variables. Let $x$ be the number of beads used for each bracelet and $y$ be the number of beads used for each necklace.
Write Equations: Based on the information given, we can write two equations. For the family, the equation is $6x + 4y = 532$ . For the friends, the equation is $4x + 7y = 697$ .
Solve System of Equations: System of equations: $\newline$ $6x + 4y = 532$ (Equation $1$ ) $\newline$ $4x + 7y = 697$ (Equation $2$ ) $\newline$ We need to solve this system for $x$ and $y$ .
Multiply Equations: Multiply Equation $1$ by $4$ and Equation $2$ by $6$ to make the coefficients of $x$ the same in both equations. $\newline$ $4(6x + 4y) = 4(532)$ $\newline$ $6(4x + 7y) = 6(697)$
Subtract Equations: After multiplying, we get: $\newline$ $24x + 16y = 2128$ (Equation $3$ ) $\newline$ $24x + 42y = 4182$ (Equation $4$ )
Solve for y: Subtract Equation $3$ from Equation $4$ to eliminate x. $\newline$ $(24x + 42y) - (24x + 16y) = 4182 - 2128$ $\newline$ $24x + 42y - 24x - 16y = 4182 - 2128$ $\newline$ $26y = 2054$
Substitute Back: Solve for $y$ . $26y = 2054$ $y = \frac{2054}{26}$ $y = 79$
Solve for x: Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use Equation $1$ . $\newline$ $6x + 4(79) = 532$
Solve for x: Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use Equation $1$ . $6x + 4(79) = 532$ Solve for $x$ . $6x + 316 = 532$ $6x = 532 - 316$ $6x = 216$ $x = \frac{216}{6}$ $x = 36$