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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ There are two mystery numbers. The sum of $3$ times the first number and $3$ times the second number is $-18$ . The sum of $2$ times the first number and $3$ times the second number is $-13$ . What are the two numbers? $\newline$ The first number is , and the second number is .

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

There are two mystery numbers. The sum of

3

times the first number and

3

times the second number is

-18

. The sum of

2

times the first number and

3

times the second number is

-13

. What are the two numbers?

\newline

The first number is ______ , and the second number is ______.

Define Equations: Let's call the first number $x$ and the second number $y$ . The first equation from the sum of $3$ times each number is $3x + 3y = -18$ .
Form System of Equations: The second equation from the sum of $2$ times the first number and $3$ times the second number is $2x + 3y = -13$ .
Eliminate Variable: Now we have a system of equations: $\newline$ $1$ ) $3x + 3y = -18$ $\newline$ $2$ ) $2x + 3y = -13$ $\newline$ We can subtract the second equation from the first to eliminate $y$ .
Solve for x: Subtracting the equations we get: $\newline$ $(3x + 3y) - (2x + 3y) = -18 - (-13)$ $\newline$ $3x - 2x + 3y - 3y = -18 + 13$ $\newline$ $x = -5$
Substitute $x$ into Equation: Now we substitute $x = -5$ into the second equation: $\newline$ $2(-5) + 3y = -13$ $\newline$ $-10 + 3y = -13$ $\newline$ $3y = -13 + 10$ $\newline$ $3y = -3$ $\newline$ $y = -1$
Final Solution: So the first number is $x = -5$ , and the second number is $y = -1$ .

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