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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMrs. Sloan is a dance teacher whose students are going to have a recital. She is comparing two companies' prices for filming the recital and producing DVDs. Joy's Productions charges $4\$4 per DVD, plus $70\$70 to come and film the recital. Lanberry Media charges $79\$79 to film, plus $3\$3 per DVD. To decide which company to use, Mrs. Sloan determines the number of DVDs that would make the two options equivalent in terms of cost. How many DVDs would that be? How much would each option cost?\newlineIf Mrs. Sloan ordered _____ DVDs, the cost would be $\$_____.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMrs. Sloan is a dance teacher whose students are going to have a recital. She is comparing two companies' prices for filming the recital and producing DVDs. Joy's Productions charges $4\$4 per DVD, plus $70\$70 to come and film the recital. Lanberry Media charges $79\$79 to film, plus $3\$3 per DVD. To decide which company to use, Mrs. Sloan determines the number of DVDs that would make the two options equivalent in terms of cost. How many DVDs would that be? How much would each option cost?\newlineIf Mrs. Sloan ordered _____ DVDs, the cost would be $\$_____.
  1. Define variables: Define variables for the number of DVDs and the total cost.\newlineLet xx represent the number of DVDs and CC represent the total cost.
  2. Write Joy's equation: Write the equation for Joy's Productions.\newlineJoy's Productions charges $4\$4 per DVD plus a $70\$70 filming fee.\newlineThe equation for Joy's Productions is: C=4x+70C = 4x + 70.
  3. Write Lanberry's equation: Write the equation for Lanberry Media.\newlineLanberry Media charges $3\$3 per DVD plus a $79\$79 filming fee.\newlineThe equation for Lanberry Media is: C=3x+79C = 3x + 79.
  4. Set up system of equations: Set up the system of equations to find the number of DVDs that would make the costs equivalent.\newline4x+70=3x+794x + 70 = 3x + 79
  5. Solve using substitution: Solve the system using substitution.\newlineSubtract 3x3x from both sides of the equation to isolate xx on one side.\newline4x+70−3x=3x+79−3x4x + 70 - 3x = 3x + 79 - 3x\newlinex+70=79x + 70 = 79
  6. Subtract to isolate x: Subtract 7070 from both sides to solve for x.\newlinex+70−70=79−70x + 70 - 70 = 79 - 70\newlinex=9x = 9
  7. Substitute x=9x = 9: Substitute x=9x = 9 back into one of the original equations to find the total cost CC. Using Joy's Productions equation: C=4x+70C = 4x + 70 C=4(9)+70C = 4(9) + 70 C=36+70C = 36 + 70 C=106C = 106
  8. Verify solution: Verify the solution by substituting x=9x = 9 into Lanberry Media's equation.\newlineC=3x+79C = 3x + 79\newlineC=3(9)+79C = 3(9) + 79\newlineC=27+79C = 27 + 79\newlineC=106C = 106

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