Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Julia is thinking of two numbers. Adding $6$ times the first number and $3$ times the second number gives a total of $18$ . Also, adding $6$ times the first number and $10$ times the second number gives $4$ . 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 elimination: word problems

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

\newline

Julia is thinking of two numbers. Adding

6

times the first number and

3

times the second number gives a total of

18

. Also, adding

6

times the first number and

10

times the second number gives

4

. What are the two numbers?

\newline

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

Define Equations: Let's denote the first number as $x$ and the second number as $y$ . The first condition given is that adding $6$ times the first number and $3$ times the second number gives a total of $18$ . This can be written as the equation $6x + 3y = 18$ .
System of Equations: The second condition is that adding $6$ times the first number and $10$ times the second number gives $4$ . This can be written as the equation $6x + 10y = 4$ .
Elimination Method: We now have a system of two equations: $\newline$ $1$ . $6x + 3y = 18$ $\newline$ $2$ . $6x + 10y = 4$ $\newline$ We will use elimination to solve this system. To eliminate $x$ , we can subtract the second equation from the first.
Subtract and Simplify: Subtracting the second equation from the first gives us: $\newline$ $(6x + 3y) - (6x + 10y) = 18 - 4$ $\newline$ This simplifies to: $\newline$ $-7y = 14$
Solve for y: We divide both sides of the equation $-7y = 14$ by $-7$ to solve for y: $\newline$ $y = \frac{14}{-7}$ $\newline$ $y = -2$
Substitute Back: Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . We'll use the first equation $6x + 3y = 18$ : $6x + 3(-2) = 18$ $6x - 6 = 18$
Isolate x: We add $6$ to both sides of the equation $6x - 6 = 18$ to isolate the term with $x$ : $\newline$ $6x = 18 + 6$ $\newline$ $6x = 24$
Final Solution: Finally, we divide both sides of the equation $6x = 24$ by $6$ to solve for $x$ : $\newline$ $x = \frac{24}{6}$ $\newline$ $x = 4$