Solve the system of equations:

[tex]
\begin{array}{l}
2.5 y+3 x=27 \\
5 x-2.5 y=5
\end{array}
[/tex]

What equation is the result of adding the two equations?
What is the solution to the system?

Add the two equations to eliminate y and get 8 x = 32 .
Solve for x : x = 8 32 ​ = 4 .
Substitute x = 4 into the first equation to solve for y : 2.5 y + 3 ( 4 ) = 27 , which gives y = 6 .
The solution to the system is ( 4 , 6 ) .

Explanation

Problem Analysis We are given a system of two linear equations:

2.5 y + 3 x = 27 5 x − 2.5 y = 5
Our goal is to find the equation that results from adding these two equations and to find the solution (x, y) to the system.

Adding the Equations First, let's add the two equations together:

( 2.5 y + 3 x ) + ( 5 x − 2.5 y ) = 27 + 5
Combining like terms, we get:
8 x = 32
So, the equation resulting from adding the two equations is:
8 x = 32

Solving for x Now, let's solve for x :

8 x = 32 x = 8 32 ​ x = 4
So, x = 4 .

Solving for y Next, we substitute the value of x into one of the original equations to solve for y . Let's use the first equation:

2.5 y + 3 x = 27 2.5 y + 3 ( 4 ) = 27 2.5 y + 12 = 27 2.5 y = 27 − 12 2.5 y = 15 y = 2.5 15 ​ y = 6
So, y = 6 .

Final Solution Therefore, the solution to the system of equations is ( x , y ) = ( 4 , 6 ) .

Examples
Systems of equations are used in various real-world applications. For instance, in economics, they can determine equilibrium prices and quantities in markets. In engineering, they can analyze circuits or structural systems. Imagine you're running a small business selling two products. By setting up a system of equations, you can determine the optimal pricing for each product to maximize your profit, considering factors like production costs and market demand. This helps in making informed business decisions.

Answered by GinnyAnswer | 2025-07-03

The result of adding the two equations is 8 x = 32 . The solution to the system is ( x , y ) = ( 4 , 6 ) .
;

Answered by Anonymous | 2025-07-04