Find the $x$ and $y$ intercepts of the equation: $-2x - 5y = 20$
The intercepts are:
$x$ intercept $=($ $\square$ ,0)
$y$ intercept $=(0$, $\square$ )
Answer (1)
To find the x -intercept, set y = 0 and solve for x : − 2 x − 5 ( 0 ) = 20 A rr x = − 10
The x -intercept is ( − 10 , 0 ) .
To find the y -intercept, set x = 0 and solve for y : − 2 ( 0 ) − 5 y = 20 A rry = − 4
The y -intercept is ( 0 , − 4 ) . The intercepts are ( − 10 , 0 ) and ( 0 , − 4 ) .
Explanation
Understanding the Problem We are given the equation − 2 x − 5 y = 20 and we need to find the x and y intercepts. The x -intercept is the point where the line crosses the x -axis, which means y = 0 . The y -intercept is the point where the line crosses the y -axis, which means x = 0 .
Finding the x-intercept To find the x -intercept, we substitute y = 0 into the equation: − 2 x − 5 ( 0 ) = 20 − 2 x = 20 x = − 2 20 x = − 10 So, the x -intercept is ( − 10 , 0 ) .
Finding the y-intercept To find the y -intercept, we substitute x = 0 into the equation: − 2 ( 0 ) − 5 y = 20 − 5 y = 20 y = − 5 20 y = − 4 So, the y -intercept is ( 0 , − 4 ) .
Final Answer Therefore, the x -intercept is ( − 10 , 0 ) and the y -intercept is ( 0 , − 4 ) .
Examples
Understanding intercepts is crucial in various real-life scenarios. For example, if you're analyzing a cost function for a business, the y-intercept represents the fixed costs (costs when production is zero), and the x-intercept (if meaningful in the context) might represent the break-even point (when costs are covered by revenue). Similarly, in physics, intercepts can represent initial conditions or equilibrium points in a system.
