Find the solution to the system of equations:
$y = \frac{3}{2}x$
$y = -\frac{3}{4}x + 9$
Solution: ____
The problem requires us to find the intersection point of two linear equations, which represents the values of \(x\) and \(y\) that satisfy both equations simultaneously:
1) \(y = \frac{3}{2}x\)
2) \(y = -\frac{3}{4}x + 9\)
Both equations express \(y\) in terms of \(x\), which makes the substitution method ideal for solving this system. We are looking for the coordinate \((x, y)\) where the two lines intersect.
Since both expressions are equal to \(y\), we can set them equal to each other to solve for \(x\):
\[\frac{3}{2}x = -\frac{3}{4}x + 9\]
To combine the \(x\) terms, we add \(\frac{3}{4}x\) to both sides:
\[\frac{3}{2}x + \frac{3}{4}x = 9\]
Finding a common denominator for the fractions on the left, we rewrite \(\frac{3}{2}x\) as \(\frac{6}{4}x\):
\[\frac{6}{4}x + \frac{3}{4}x = 9\]
\[\frac{9}{4}x = 9\]
Now, we solve for \(x\) by multiplying both sides by the reciprocal, which is \(\frac{4}{9}\):
\[x = 9 \cdot \frac{4}{9} = 4\]
Now that we have the value of \(x = 4\), we substitute it back into the first equation to find \(y\):
\[y = \frac{3}{2}(4)\]
\[y = \frac{3}{2} \cdot 4 = \frac{12}{2} = 6\]
So, the solution to the system is \(x = 4\) and \(y = 6\). To verify, we substitute these values into the second equation:
\[6 = -\frac{3}{4}(4) + 9\]
\[6 = -3 + 9\]
\[6 = 6\]
The equality holds, confirming that \((4, 6)\) is indeed the intersection point. The final answer is \(\boxed{(4, 6)}\).
In real-life scenarios, solving systems of linear equations is used for finding the "break-even point" in business. For example, if one function represents the cost of production and another represents the total revenue from sales, the intersection point shows exactly how many units must be produced and sold for the business to cover all costs and start making a profit. This mathematical technique allows planners to optimize resources and make informed financial decisions.