Use substitution to solve the system:
$\begin{array}{l}
y=3.2 x+1.9 \\
2.3 x+2 y=17.72
\end{array}$
We are given a system of two linear equations with two variables, \(x\) and \(y\):
\[ \begin{cases} y = 3.2x + 1.9 \ 2.3x + 2y = 17.72 \end{cases} \]
Our objective is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously using the substitution method. The first equation already provides an expression for \(y\) in terms of \(x\), which makes substitution the most direct approach.
The first equation is \(y = 3.2x + 1.9\). We can substitute this expression for \(y\) into the second equation, \(2.3x + 2y = 17.72\):
\[2.3x + 2(3.2x + 1.9) = 17.72\]
Now, we distribute the constant \(2\) inside the parentheses:
\[2.3x + (2 \times 3.2x) + (2 \times 1.9) = 17.72\]
\[2.3x + 6.4x + 3.8 = 17.72\]
Combining the like terms involving \(x\):
\[(2.3 + 6.4)x + 3.8 = 17.72\]
\[8.7x + 3.8 = 17.72\]
To solve for \(x\), we first isolate the term with \(x\) by subtracting \(3.8\) from both sides of the equation:
\[8.7x = 17.72 - 3.8\]
\[8.7x = 13.92\]
Next, divide both sides by \(8.7\):
\[x = \frac{13.92}{8.7} = 1.6\]
With \(x = 1.6\), we can find \(y\) by substituting this value back into the first equation:
\[y = 3.2(1.6) + 1.9\]
\[y = 5.12 + 1.9\]
\[y = 7.02\]
Thus, the solution to the system is \(x = 1.6\) and \(y = 7.02\).
This substitution method is widely used in economics and engineering to find equilibrium points in systems. For example, if you have a demand curve \(y = 3.2x + 1.9\) and a supply curve \(2.3x + 2y = 17.72\) where \(x\) represents price and \(y\) represents quantity, solving this system tells you the market equilibrium price and quantity. Mastering this skill allows you to analyze how different variables interact in real-world models.