Solve the system of equations:
$y=3 x+4$
$y=-\frac{1}{2} x+1$
The given problem is a system of two linear equations in two variables, \(x\) and \(y\):
\[y = 3x + 4\]
\[y = -\frac{1}{2}x + 1\]
These equations represent two straight lines in a 2D coordinate system. Solving the system means finding the point \((x, y)\) where these two lines intersect. Since both equations are already solved for \(y\), we can use the method of substitution by setting the expressions for \(y\) equal to each other.
To find the value of \(x\) at the intersection point, we set the right-hand sides of the two equations equal to each other:
\[3x + 4 = -\frac{1}{2}x + 1\]
To solve for \(x\), we first clear the fraction by multiplying every term on both sides by 2:
\[2 \times (3x + 4) = 2 \times \left(-\frac{1}{2}x + 1\right)\]
\[6x + 8 = -x + 2\]
Now, add \(x\) to both sides and subtract 8 from both sides to isolate the \(x\) terms:
\[6x + x = 2 - 8\]
\[7x = -6\]
Dividing by 7, we find:
\[x = -\frac{6}{7}\]
Now that we have the value of \(x\), we substitute it back into one of the original equations to find the corresponding value of \(y\). Using the first equation \(y = 3x + 4\):
\[y = 3\left(-\frac{6}{7}\right) + 4\]
\[y = -\frac{18}{7} + 4\]
To add these, convert 4 into a fraction with a denominator of 7: \(4 = \frac{28}{7}\).
\[y = -\frac{18}{7} + \frac{28}{7} = \frac{10}{7}\]
We can verify this by substituting \(x = -\frac{6}{7}\) into the second equation:
\[y = -\frac{1}{2}\left(-\frac{6}{7}\right) + 1 = \frac{3}{7} + 1 = \frac{3}{7} + \frac{7}{7} = \frac{10}{7}\]
Since both equations yield the same value for \(y\), the solution is verified.
The intersection point of the two lines represented by the system of equations is \(\left(-\frac{6}{7}, \frac{10}{7}\right)\). This is the only point that satisfies both equations simultaneously. The final answer is \(\boxed{\left(-\frac{6}{7}, \frac{10}{7}\right)}\).
Systems of linear equations are used extensively in economics to find the equilibrium point where supply and demand curves intersect. For example, if you have a supply curve representing the cost of producing goods and a demand curve representing how much customers are willing to pay, the point of intersection tells you the price and quantity that will clear the market. Understanding this allows businesses and policymakers to predict outcomes and optimize resources.