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Twice one number added to three times another number equals -3 . If the first number is tripled and subtracted from 8 and the result is divided by 2 , then the second number is obtained. Find the two numbers.

Twice one number added to three times another number equals −3-3 . If the first number is tripled and subtracted from 88 and the result is divided by 22 , then the second number is obtained. Find the two numbers.

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Full solution

Q. Twice one number added to three times another number equals −3-3 . If the first number is tripled and subtracted from 88 and the result is divided by 22 , then the second number is obtained. Find the two numbers.
  1. Translate Equations: Let's denote the first number as xx and the second number as yy. The problem gives us two equations based on the description:\newline11. Twice one number (xx) added to three times another number (yy) equals −3-3.\newline22. If the first number (xx) is tripled and subtracted from 88, and the result is divided by 22, then the second number (yy) is obtained.\newlineWe can translate these descriptions into algebraic equations:\newline11. 2x+3y=−32x + 3y = -3\newline22. yy00\newlineNow we have a system of two equations with two variables.
  2. Solve for y: Let's solve the second equation for y to make it easier to substitute into the first equation:\newline(8−3x)/2=y(8 - 3x) / 2 = y\newlineMultiply both sides by 22 to get rid of the fraction:\newline8−3x=2y8 - 3x = 2y\newlineNow we can express y in terms of x:\newliney=(8−3x)/2y = (8 - 3x) / 2
  3. Substitute into First Equation: Next, we substitute the expression for yy into the first equation:\newline2x+3((8−3x)/2)=−32x + 3((8 - 3x) / 2) = -3\newlineNow we need to distribute the 33 inside the parentheses:\newline2x+(3×8)/2−(3×3x)/2=−32x + (3 \times 8)/2 - (3 \times 3x)/2 = -3\newlineSimplify the equation:\newline2x+12−(9x/2)=−32x + 12 - (9x/2) = -3
  4. Combine and Solve for x: Now, let's combine like terms and solve for x:\newlineTo combine the terms, we need a common denominator for x terms, which is 22:\newline(4x/2)−(9x/2)+12=−3(4x/2) - (9x/2) + 12 = -3\newlineCombine the x terms:\newline(−5x/2)+12=−3(-5x/2) + 12 = -3\newlineNow, subtract 1212 from both sides to isolate the x term:\newline(−5x/2)=−3−12(-5x/2) = -3 - 12\newline(−5x/2)=−15(-5x/2) = -15
  5. Find xx: Next, we multiply both sides by −25-\frac{2}{5} to solve for xx:
    x=(−15)×(−25)x = (-15) \times (-\frac{2}{5})
    x=305x = \frac{30}{5}
    x=6x = 6
    We have found the value of the first number, xx.
  6. Substitute xx to Find yy: Now that we have xx, we can find yy by substituting xx back into the equation we found for yy:
    y=8−3x2y = \frac{8 - 3x}{2}
    y=8−3(6)2y = \frac{8 - 3(6)}{2}
    y=8−182y = \frac{8 - 18}{2}
    y=−102y = \frac{-10}{2}
    yy00
    We have found the value of the second number, yy.

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