Solve the simultaneous equations:
$\begin{array}{l}
y=x+1 \text{ and } y=3-x.\\
x=\quad, \quad y=
\end{array}$
To solve the system of equations, we are given two linear equations:
1) \(y = x + 1\)
2) \(y = 3 - x\)
Both equations express \(y\) in terms of \(x\). Since both expressions represent the same variable \(y\), we can set them equal to each other to solve for \(x\).
Equating the two expressions for \(y\):
\[x + 1 = 3 - x\]
To isolate \(x\), we first add \(x\) to both sides of the equation:
\[x + x + 1 = 3\]
\[2x + 1 = 3\]
Next, subtract \(1\) from both sides:
\[2x = 3 - 1\]
\[2x = 2\]
Finally, divide by \(2\):
\[x = 1\]
Thus, the value of \(x\) is \(1\).
Now that we have the value \(x = 1\), we can substitute it into one of the original equations to find the value of \(y\). Using the first equation:
\[y = x + 1\]
\[y = 1 + 1\]
\[y = 2\]
To ensure accuracy, we verify this with the second equation:
\[y = 3 - x\]
\[y = 3 - 1\]
\[y = 2\]
Since both equations yield \(y = 2\), the solution is confirmed. The values that satisfy the system are \(x = 1\) and \(y = 2\).
This method of solving a system of linear equations is commonly used in economics to find the 'equilibrium price' where supply and demand intersect. For example, if the demand function for a product is \(P = 3 - Q\) and the supply function is \(P = Q + 1\), where \(P\) is price and \(Q\) is quantity, finding the intersection point tells the producer the price and quantity that balances the market. Here, setting \(Q + 1 = 3 - Q\) results in \(2Q = 2\), so \(Q = 1\), and the equilibrium price is \(P = 1 + 1 = 2\).