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.

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Q. 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.

Translate Equations: Let's denote the first number as $x$ and the second number as $y$ . The problem gives us two equations based on the description: $\newline$ $1$ . Twice one number ( $x$ ) added to three times another number ( $y$ ) equals $-3$ . $\newline$ $2$ . If the first number ( $x$ ) is tripled and subtracted from $8$ , and the result is divided by $2$ , then the second number ( $y$ ) is obtained. $\newline$ We can translate these descriptions into algebraic equations: $\newline$ $1$ . $2x + 3y = -3$ $\newline$ $2$ . $y$ $0$ $\newline$ Now we have a system of two equations with two variables.
Solve for y: Let's solve the second equation for y to make it easier to substitute into the first equation: $\newline$ $(8 - 3x) / 2 = y$ $\newline$ Multiply both sides by $2$ to get rid of the fraction: $\newline$ $8 - 3x = 2y$ $\newline$ Now we can express y in terms of x: $\newline$ $y = (8 - 3x) / 2$
Substitute into First Equation: Next, we substitute the expression for $y$ into the first equation: $\newline$ $2x + 3((8 - 3x) / 2) = -3$ $\newline$ Now we need to distribute the $3$ inside the parentheses: $\newline$ $2x + (3 \times 8)/2 - (3 \times 3x)/2 = -3$ $\newline$ Simplify the equation: $\newline$ $2x + 12 - (9x/2) = -3$
Combine and Solve for x: Now, let's combine like terms and solve for x: $\newline$ To combine the terms, we need a common denominator for x terms, which is $2$ : $\newline$ $(4x/2) - (9x/2) + 12 = -3$ $\newline$ Combine the x terms: $\newline$ $(-5x/2) + 12 = -3$ $\newline$ Now, subtract $12$ from both sides to isolate the x term: $\newline$ $(-5x/2) = -3 - 12$ $\newline$ $(-5x/2) = -15$
Find $x$ : Next, we multiply both sides by $-\frac{2}{5}$ to solve for $x$ :
$x = (-15) \times (-\frac{2}{5})$
$x = \frac{30}{5}$
$x = 6$
We have found the value of the first number, $x$ .
Substitute $x$ to Find $y$ : Now that we have $x$ , we can find $y$ by substituting $x$ back into the equation we found for $y$ :
$y = \frac{8 - 3x}{2}$
$y = \frac{8 - 3(6)}{2}$
$y = \frac{8 - 18}{2}$
$y = \frac{-10}{2}$
$y$ $0$
We have found the value of the second number, $y$ .