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A teacher purchased 300300 colored pencils for an upcoming project. The pencils she ordered come in packs of 2020 and packs of 3030. She ordered 1414 packs in all. Write a system of equations to find the number of 2020-pencil packs (x)(x) and 3030-pencil packs (y)(y) the art teacher ordered.

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Q. A teacher purchased 300300 colored pencils for an upcoming project. The pencils she ordered come in packs of 2020 and packs of 3030. She ordered 1414 packs in all. Write a system of equations to find the number of 2020-pencil packs (x)(x) and 3030-pencil packs (y)(y) the art teacher ordered.
  1. Define Variables and Equations: Define the variables and write the equations based on the given information.\newlineThe teacher ordered a total of 1414 packs, which can be represented by the equation:\newlinex+y=14x + y = 14\newlineThis equation represents the total number of packs ordered.
  2. Write Total Pencils Equation: Write the second equation based on the total number of pencils.\newlineThe teacher ordered 300300 colored pencils in total. Since packs come in 2020s and 3030s, we can represent this with the equation:\newline20x+30y=30020x + 30y = 300\newlineThis equation represents the total number of pencils ordered.
  3. Solve Using Elimination: Solve the system of equations using substitution or elimination.\newlineWe can use the elimination method by multiplying the first equation by −20-20 to eliminate xx:\newline−20(x+y)=−20(14)-20(x + y) = -20(14)\newline−20x−20y=−280-20x - 20y = -280\newlineNow we have the system:\newline−20x−20y=−280-20x - 20y = -280\newline20x+30y=30020x + 30y = 300
  4. Add Equations to Eliminate xx: Add the two equations together to eliminate xx.(−20x−20y)+(20x+30y)=−280+300(-20x - 20y) + (20x + 30y) = -280 + 300The xx terms cancel out, and we are left with:10y=2010y = 20
  5. Solve for y: Solve for y.\newlineDivide both sides of the equation by 1010 to find the value of y:\newline10y10=2010\frac{10y}{10} = \frac{20}{10}\newliney=2y = 2\newlineThis means the teacher ordered 22 packs of 3030-pencil packs.
  6. Substitute to Find x: Substitute the value of yy back into one of the original equations to solve for xx. Using the first equation: x+y=14x + y = 14 x+2=14x + 2 = 14 Subtract 22 from both sides to solve for xx: x=14−2x = 14 - 2 x=12x = 12 This means the teacher ordered 1212 packs of 2020-pencil packs.

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