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Twice one number added to thee limes another number equals 3 . If the firat 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 thee limes another number equals 33 . If the firat 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 thee limes another number equals 33 . If the firat 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: Translate the word problem into a system of equations.\newlineThe first sentence ＂Twice one number added to three times another number equals 33＂ can be translated to the equation 2x+3y=32x + 3y = 3, where xx is the first number and yy is the second number.\newlineThe second sentence ＂If the first number is tripled and subtracted from 88 and the result is divided by 22, then the second number is obtained＂ can be translated to the equation (8−3x)/2=y(8 - 3x) / 2 = y.
  2. Solve for y: Solve the second equation for y to get y in terms of x.\newlineStarting with (8−3x)/2=y(8 - 3x) / 2 = y, we can multiply both sides by 22 to get rid of the fraction:\newline2×((8−3x)/2)=2×y2 \times ((8 - 3x) / 2) = 2 \times y\newline8−3x=2y8 - 3x = 2y\newlineNow we can express y in terms of x:\newliney=(8−3x)/2y = (8 - 3x) / 2
  3. Substitute and Simplify: Substitute the expression for yy from Step 22 into the first equation.\newlineWe have the first equation 2x+3y=32x + 3y = 3 and y=8−3x2y = \frac{8 - 3x}{2} from Step 22. Substituting yy into the first equation gives us:\newline2x+3×(8−3x2)=32x + 3 \times \left(\frac{8 - 3x}{2}\right) = 3
  4. Solve for x: Solve the resulting equation for x.\newlineFirst, distribute the 33 on the left side of the equation:\newline2x+(3×8)/2−(3×3x)/2=32x + (3 \times 8)/2 - (3 \times 3x)/2 = 3\newline2x+12−(9x)/2=32x + 12 - (9x)/2 = 3\newlineNow, combine like terms and solve for x:\newline(4x)/2−(9x)/2=3−12(4x)/2 - (9x)/2 = 3 - 12\newline(−5x)/2=−9(-5x)/2 = -9\newlineMultiply both sides by −2/5-2/5 to solve for x:\newlinex=(−9)×(−2/5)x = (-9) \times (-2/5)\newlinex=18/5x = 18/5\newlinex=3.6x = 3.6
  5. Substitute for y: Substitute the value of xx back into the expression for yy from Step 22 to find the value of yy.
    y=8−3x2y = \frac{8 - 3x}{2}
    y=8−3×3.62y = \frac{8 - 3 \times 3.6}{2}
    y=8−10.82y = \frac{8 - 10.8}{2}
    y=−2.82y = \frac{-2.8}{2}
    y=−1.4y = -1.4
  6. Check Solution: Check the solution by substituting both xx and yy into the original equations.\newlineFirst equation: 2x+3y=32x + 3y = 3\newline2(3.6)+3(−1.4)=32(3.6) + 3(-1.4) = 3\newline7.2−4.2=37.2 - 4.2 = 3\newline3=33 = 3 (True)\newlineSecond equation: (8−3x)/2=y(8 - 3x) / 2 = y\newline(8−3(3.6))/2=−1.4(8 - 3(3.6)) / 2 = -1.4\newline(8−10.8)/2=−1.4(8 - 10.8) / 2 = -1.4\newline−2.8/2=−1.4-2.8 / 2 = -1.4\newlineyy00 (True)\newlineBoth equations are satisfied with yy11 and yy22.

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