Find the $y$-intercept.
$\begin{array}{l}
y=\frac{3 x+30}{x-10} \\
(0,[?])
\end{array}$
Answer (2)
To find the y -intercept, set x = 0 in the equation.
Substitute x = 0 into the equation: y = 0 − 10 3 ( 0 ) + 30 .
Simplify the expression: y = − 10 30 = − 3 .
The y -intercept is − 3 .
Explanation
Understanding the Problem We are given the equation y = x − 10 3 x + 30 and asked to find the y -intercept. The y -intercept is the point where the graph of the equation intersects the y -axis. This occurs when x = 0 .
Substituting x=0 To find the y -intercept, we substitute x = 0 into the equation: y = 0 − 10 3 ( 0 ) + 30
Simplifying the Expression Now, we simplify the expression to find the value of y :
y = − 10 0 + 30 = − 10 30 = − 3
Finding the y-intercept Therefore, the y -intercept is the point ( 0 , − 3 ) .
Examples
Understanding y-intercepts is crucial in many real-world applications. For example, in business, if you graph the cost of producing items versus the number of items produced, the y-intercept represents the fixed costs (the costs you have even if you produce nothing). Similarly, in physics, if you graph the position of an object over time, the y-intercept represents the object's initial position. Knowing how to find the y-intercept allows you to quickly determine these important initial values or fixed costs, providing a starting point for further analysis.
To find the y -intercept of the function y = x − 10 3 x + 30 , we substitute x = 0 into the equation, resulting in y = − 3 . Thus, the y -intercept is the point ( 0 , − 3 ) .
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