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.

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Q. 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.

Translate Equations: Translate the word problem into a system of equations. $\newline$ The first sentence ＂Twice one number added to three times another number equals $3$ ＂ can be translated to the equation $2x + 3y = 3$ , where $x$ is the first number and $y$ is the second number. $\newline$ The second sentence ＂If the first number is tripled and subtracted from $8$ and the result is divided by $2$ , then the second number is obtained＂ can be translated to the equation $(8 - 3x) / 2 = y$ .
Solve for y: Solve the second equation for y to get y in terms of x. $\newline$ Starting with $(8 - 3x) / 2 = y$ , we can multiply both sides by $2$ to get rid of the fraction: $\newline$ $2 \times ((8 - 3x) / 2) = 2 \times y$ $\newline$ $8 - 3x = 2y$ $\newline$ Now we can express y in terms of x: $\newline$ $y = (8 - 3x) / 2$
Substitute and Simplify: Substitute the expression for $y$ from Step $2$ into the first equation. $\newline$ We have the first equation $2x + 3y = 3$ and $y = \frac{8 - 3x}{2}$ from Step $2$ . Substituting $y$ into the first equation gives us: $\newline$ $2x + 3 \times \left(\frac{8 - 3x}{2}\right) = 3$
Solve for x: Solve the resulting equation for x. $\newline$ First, distribute the $3$ on the left side of the equation: $\newline$ $2x + (3 \times 8)/2 - (3 \times 3x)/2 = 3$ $\newline$ $2x + 12 - (9x)/2 = 3$ $\newline$ Now, combine like terms and solve for x: $\newline$ $(4x)/2 - (9x)/2 = 3 - 12$ $\newline$ $(-5x)/2 = -9$ $\newline$ Multiply both sides by $-2/5$ to solve for x: $\newline$ $x = (-9) \times (-2/5)$ $\newline$ $x = 18/5$ $\newline$ $x = 3.6$
Substitute for y: Substitute the value of $x$ back into the expression for $y$ from Step $2$ to find the value of $y$ .
$y = \frac{8 - 3x}{2}$
$y = \frac{8 - 3 \times 3.6}{2}$
$y = \frac{8 - 10.8}{2}$
$y = \frac{-2.8}{2}$
$y = -1.4$
Check Solution: Check the solution by substituting both $x$ and $y$ into the original equations. $\newline$ First equation: $2x + 3y = 3$ $\newline$ $2(3.6) + 3(-1.4) = 3$ $\newline$ $7.2 - 4.2 = 3$ $\newline$ $3 = 3$ (True) $\newline$ Second equation: $(8 - 3x) / 2 = y$ $\newline$ $(8 - 3(3.6)) / 2 = -1.4$ $\newline$ $(8 - 10.8) / 2 = -1.4$ $\newline$ $-2.8 / 2 = -1.4$ $\newline$ $y$ $0$ (True) $\newline$ Both equations are satisfied with $y$ $1$ and $y$ $2$ .