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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineValentina is creating beaded jewelry to give to her family and friends. For her family, she assembled 66 bracelets and 44 necklaces, using a total of 532532 beads. For her friends, she assembled 44 bracelets and 77 necklaces, using a total of 697697 beads. Assuming she uses a consistent number of beads for every bracelet and necklace, how many beads is she using for each?\newlineValentina uses _____ beads for each bracelet and _____ beads for each necklace.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineValentina is creating beaded jewelry to give to her family and friends. For her family, she assembled 66 bracelets and 44 necklaces, using a total of 532532 beads. For her friends, she assembled 44 bracelets and 77 necklaces, using a total of 697697 beads. Assuming she uses a consistent number of beads for every bracelet and necklace, how many beads is she using for each?\newlineValentina uses _____ beads for each bracelet and _____ beads for each necklace.
  1. Define Variables: Let's define the variables. Let xx be the number of beads used for each bracelet and yy be the number of beads used for each necklace.
  2. Write Equations: Based on the information given, we can write two equations. For the family, the equation is 6x+4y=5326x + 4y = 532. For the friends, the equation is 4x+7y=6974x + 7y = 697.
  3. Solve System of Equations: System of equations:\newline6x+4y=5326x + 4y = 532 (Equation 11)\newline4x+7y=6974x + 7y = 697 (Equation 22)\newlineWe need to solve this system for xx and yy.
  4. Multiply Equations: Multiply Equation 11 by 44 and Equation 22 by 66 to make the coefficients of xx the same in both equations.\newline4(6x+4y)=4(532)4(6x + 4y) = 4(532)\newline6(4x+7y)=6(697)6(4x + 7y) = 6(697)
  5. Subtract Equations: After multiplying, we get:\newline24x+16y=212824x + 16y = 2128 (Equation 33)\newline24x+42y=418224x + 42y = 4182 (Equation 44)
  6. Solve for y: Subtract Equation 33 from Equation 44 to eliminate x.\newline(24x+42y)−(24x+16y)=4182−2128(24x + 42y) - (24x + 16y) = 4182 - 2128\newline24x+42y−24x−16y=4182−212824x + 42y - 24x - 16y = 4182 - 2128\newline26y=205426y = 2054
  7. Substitute Back: Solve for yy.26y=205426y = 2054y=205426y = \frac{2054}{26}y=79y = 79
  8. Solve for x: Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. Let's use Equation 11.\newline6x+4(79)=5326x + 4(79) = 532
  9. Solve for x: Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. Let's use Equation 11.6x+4(79)=5326x + 4(79) = 532Solve for xx.6x+316=5326x + 316 = 5326x=532−3166x = 532 - 3166x=2166x = 216x=2166x = \frac{216}{6}x=36x = 36

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