Solve the system by substitution.

[tex]\begin{array}{c}
9 x+3 y=-30 \\
-5 x=y
\end{array}[/tex]

Substitute y = − 5 x into the first equation: 9 x + 3 ( − 5 x ) = − 30 .
Simplify and solve for x : 9 x − 15 x = − 30 ⇒ − 6 x = − 30 ⇒ x = 5 .
Substitute x = 5 into y = − 5 x to find y : y = − 5 ( 5 ) = − 25 .
The solution is ( 5 , − 25 ) , so the answer is ( 5 , − 25 ) ​ .

Explanation

Understanding the Problem We are given a system of two equations with two variables x and y. The goal is to solve this system using the substitution method. The equations are:

9 x + 3 y = − 30 − 5 x = y

Substitution We will substitute the expression for y from the second equation into the first equation. The second equation is y = − 5 x . Substituting this into the first equation, we get:

9 x + 3 ( − 5 x ) = − 30

Simplifying the Equation Now, we simplify the equation:

9 x − 15 x = − 30

Combining Like Terms Combining like terms, we have:

− 6 x = − 30

Solving for x Now, we solve for x by dividing both sides by -6:

x = − 6 − 30 ​ = 5

Solving for y Next, we substitute the value of x back into the second equation to find y :

y = − 5 ( 5 ) = − 25

The Solution Therefore, the solution to the system of equations is x = 5 and y = − 25 . We write the solution as an ordered pair ( x , y ) .

Final Answer The solution to the system of equations is ( 5 , − 25 ) .


Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business, modeling supply and demand in economics, and solving for unknown variables in physics and engineering. For example, if you are running a business, you might use a system of equations to determine how many products you need to sell to cover your costs and start making a profit. Similarly, engineers use systems of equations to analyze the forces acting on a structure and ensure its stability.

Answered by GinnyAnswer | 2025-07-03

To solve the system of equations 9 x + 3 y = − 30 and − 5 x = y , we substitute − 5 x into the first equation, simplify, and find x = 5 and y = − 25 . The final solution is ( 5 , − 25 ) .
;

Answered by Anonymous | 2025-07-04