Solve the simultaneous equations:
$\begin{array}{l}
y = x + 1 \text{ and } x = 2.
\end{array}$
$x = \quad, y = \quad$
We are given a system of two linear equations in two variables, \(x\) and \(y\):
Our objective is to find the pair of values \((x, y)\) that satisfy both equations simultaneously. Since the second equation explicitly provides the value of \(x\), we can use the method of substitution to find the corresponding value of \(y\).
We substitute the value \(x = 2\) from the second equation into the first equation:
\[y = x + 1\]
Substituting \(x = 2\):
\[y = 2 + 1\]
\[y = 3\]
Thus, we have found that when \(x = 2\), the corresponding value of \(y\) is \(3\).
Having solved for both variables, we have determined that the values satisfying the system of equations are \(x = 2\) and \(y = 3\).
The solution to the system is:
\[x = 2, y = 3\]
This can also be expressed as the ordered pair \((2, 3)\).
This type of problem is fundamental in many fields, including economics and engineering. For example, if you are modeling a simple supply and demand situation, the intersection point of two linear functions (the supply curve and the demand curve) represents the equilibrium point. Solving the system of equations, like the one we just worked through, allows you to determine exactly where that equilibrium occurs, enabling businesses to set optimal prices for goods.