Solve the system of equations:
$y=3 x-4$
$y=-\frac{1}{2} x+10$
Solution: ____
Check:
We are given a system of two linear equations in two variables, \(x\) and \(y\):
1) \(y = 3x - 4\)
2) \(y = -\frac{1}{2}x + 10\)
The goal is to find the intersection point \((x, y)\) that satisfies both equations simultaneously. Since both equations are expressed in terms of \(y\), we can set the expressions for \(y\) equal to each other to solve for \(x\).
To find the intersection, we set the right-hand sides of the two equations equal:
\[3x - 4 = -\frac{1}{2}x + 10\]
To solve for \(x\), we first add \(\frac{1}{2}x\) to both sides of the equation:
\[3x + \frac{1}{2}x - 4 = 10\]
Combine the \(x\)-terms. Since \(3 = \frac{6}{2}\), we have \(\frac{6}{2}x + \frac{1}{2}x = \frac{7}{2}x\):
\[\frac{7}{2}x - 4 = 10\]
Next, add 4 to both sides of the equation:
\[\frac{7}{2}x = 14\]
To isolate \(x\), multiply both sides by the reciprocal \(\frac{2}{7}\):
\[x = 14 \times \frac{2}{7} = \frac{28}{7} = 4\]
So, the \(x\)-coordinate of the intersection point is \(4\).
Now that we have the \(x\)-coordinate, we substitute \(x = 4\) into the first equation to find the corresponding \(y\)-value:
\[y = 3(4) - 4\]
\[y = 12 - 4\]
\[y = 8\]
We can verify this by substituting \(x = 4\) into the second equation:
\[y = -\frac{1}{2}(4) + 10\]
\[y = -2 + 10 = 8\]
Since both equations yield the same value for \(y\), the intersection point of the two lines is \((4, 8)\).
In economics, finding the intersection point of two linear functions is a common task known as finding the equilibrium point. For example, if one line represents the supply curve of a product (where producers are willing to sell at a certain price) and the other line represents the demand curve (where consumers are willing to buy at a certain price), the intersection point tells us the market equilibrium price and the quantity that will be traded. By solving the system, we find the exact price and quantity where supply meets demand, ensuring market efficiency.