Solve the simultaneous equations:
$\begin{array}{l}
5x + 8y = 4 \\
x + 3y = -2
\end{array}$
We are given a system of two linear equations with two variables, \(x\) and \(y\):
1) \(5x + 8y = 4\)
2) \(x + 3y = -2\)
To find the values of \(x\) and \(y\) that satisfy both equations, we can use the method of substitution or elimination. Here, we will use the substitution method by expressing one variable in terms of the other from one of the equations.
From the second equation, \(x + 3y = -2\), we can isolate \(x\) by subtracting \(3y\) from both sides:
\(x = -2 - 3y\)
Now, we substitute this expression for \(x\) into the first equation, \(5x + 8y = 4\):
\(5(-2 - 3y) + 8y = 4\)
Expanding the expression:
\(-10 - 15y + 8y = 4\)
Combine the terms containing \(y\):
\(-10 - 7y = 4\)
Add 10 to both sides:
\(-7y = 14\)
Divide both sides by \(-7\):
\(y = -2\)
Now that we have found the value of \(y\), we substitute it back into the expression we found for \(x\):
\(x = -2 - 3y\)
\(x = -2 - 3(-2)\)
\(x = -2 + 6\)
\(x = 4\)
Thus, the values that satisfy both equations are \(x = 4\) and \(y = -2\). We can check this by plugging the values back into the original equations:
Equation 1: \(5(4) + 8(-2) = 20 - 16 = 4\) (Correct)
Equation 2: \(4 + 3(-2) = 4 - 6 = -2\) (Correct)
Systems of linear equations are widely used in economics to determine equilibrium points where supply equals demand. For example, if the supply function is represented by one linear equation and the demand function by another, finding their intersection helps business analysts determine the optimal price and quantity for a product. In this specific case, the variables \(x\) and \(y\) could represent the quantity and price, respectively, allowing a manager to balance their inventory and pricing strategy effectively.