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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineJulia is thinking of two numbers. Adding 66 times the first number and 33 times the second number gives a total of 1818. Also, adding 66 times the first number and 1010 times the second number gives 44. What are the two numbers?\newlineThe first number is _____, and the second number is _____.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineJulia is thinking of two numbers. Adding 66 times the first number and 33 times the second number gives a total of 1818. Also, adding 66 times the first number and 1010 times the second number gives 44. What are the two numbers?\newlineThe first number is _____, and the second number is _____.
  1. Define Equations: Let's denote the first number as xx and the second number as yy. The first condition given is that adding 66 times the first number and 33 times the second number gives a total of 1818. This can be written as the equation 6x+3y=186x + 3y = 18.
  2. System of Equations: The second condition is that adding 66 times the first number and 1010 times the second number gives 44. This can be written as the equation 6x+10y=46x + 10y = 4.
  3. Elimination Method: We now have a system of two equations:\newline11. 6x+3y=186x + 3y = 18\newline22. 6x+10y=46x + 10y = 4\newlineWe will use elimination to solve this system. To eliminate xx, we can subtract the second equation from the first.
  4. Subtract and Simplify: Subtracting the second equation from the first gives us:\newline(6x+3y)−(6x+10y)=18−4(6x + 3y) - (6x + 10y) = 18 - 4\newlineThis simplifies to:\newline−7y=14-7y = 14
  5. Solve for y: We divide both sides of the equation −7y=14-7y = 14 by −7-7 to solve for y:\newliney=14−7y = \frac{14}{-7}\newliney=−2y = -2
  6. Substitute Back: Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. We'll use the first equation 6x+3y=186x + 3y = 18:6x+3(−2)=186x + 3(-2) = 186x−6=186x - 6 = 18
  7. Isolate x: We add 66 to both sides of the equation 6x−6=186x - 6 = 18 to isolate the term with xx: \newline6x=18+66x = 18 + 6\newline6x=246x = 24
  8. Final Solution: Finally, we divide both sides of the equation 6x=246x = 24 by 66 to solve for xx: \newlinex=246x = \frac{24}{6}\newlinex=4x = 4

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