Solve the system of equations:
$\begin{array}{l}y=3.2 x+1.9 \\ 2.3 x+2 y=17.72\end{array}$
We are given a system of two linear equations in two variables, \(x\) and \(y\):
The first equation already expresses \(y\) in terms of \(x\), which makes the substitution method ideal for solving this system. Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.
We substitute the expression for \(y\) from the first equation into the second equation:
\[2.3x + 2(3.2x + 1.9) = 17.72\]
Next, we expand the terms inside the parentheses:
\[2.3x + 6.4x + 3.8 = 17.72\]
Combine the terms containing \(x\):
\[(2.3 + 6.4)x + 3.8 = 17.72\]
\[8.7x + 3.8 = 17.72\]
Now, isolate the term with \(x\) by subtracting 3.8 from both sides of the equation:
\[8.7x = 17.72 - 3.8\]
\[8.7x = 13.92\]
Finally, solve for \(x\) by dividing both sides by 8.7:
\[x = \frac{13.92}{8.7}\]
\[x = 1.6\]
With the value of \(x = 1.6\) determined, we substitute this back into the first equation to find the corresponding value of \(y\):
\[y = 3.2(1.6) + 1.9\]
\[y = 5.12 + 1.9\]
\[y = 7.02\]
Thus, the solution to the system is \(x = 1.6\) and \(y = 7.02\). The final result is \(\boxed{x=1.6, y=7.02}\).
Systems of equations like this are incredibly useful in economics and logistics. For instance, imagine a company selling two types of products. The first equation could represent the total cost of production, while the second could represent the total revenue for a specific sales volume. By solving the system, the business owner can find the exact quantity of products needed to reach a specific financial goal. Understanding how to solve these problems helps in making data-driven decisions in everyday business and personal finance management.