Find the $x$-intercept.
$\begin{array}{c}
y=\frac{3 x+30}{x-10} \
([?], 0)
\end{array}$
Answer (2)
Set y = 0 in the equation y = x − 10 3 x + 30 .
Solve the equation 0 = x − 10 3 x + 30 for x .
Set the numerator equal to zero: 3 x + 30 = 0 .
Solve for x : x = − 10 . The x -intercept is − 10 .
Explanation
Understanding the Problem We are given the equation y = x − 10 3 x + 30 and asked to find the x -intercept. The x -intercept is the point where the graph of the equation intersects the x -axis, which means we need to find the value of x when y = 0 .
Setting y = 0 To find the x -intercept, we set y = 0 in the equation: 0 = x − 10 3 x + 30
Solving for x A fraction is equal to zero if and only if its numerator is equal to zero. Therefore, we need to solve the equation: 3 x + 30 = 0
Finding the x-intercept Subtract 30 from both sides of the equation: 3 x = − 30 Divide both sides by 3: x = 3 − 30 = − 10 Thus, the x -intercept is x = − 10 .
Stating the Answer The x -intercept is the point ( − 10 , 0 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in business, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. Similarly, in physics, it can represent the point where a projectile lands. Knowing how to find x-intercepts allows us to analyze and interpret data effectively in different scenarios.
The x -intercept of the given equation is found by setting y to zero and solving for x . This gives us x = − 10 , so the x -intercept is the point ( − 10 , 0 ) . Therefore, the answer is that the x -intercept is ( − 10 , 0 ) .
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