Find the $x$ and $y$ intercepts of this linear equation.
$-3 x+11 y=66$
x-intercept ([?], $\square$ )
y-intercept ($\square$, $\square$ )
Answer (1)
To find the x-intercept, set y = 0 and solve for x : − 3 x + 11 ( 0 ) = 66 A rr x = − 22 . Thus, the x-intercept is ( − 22 , 0 ) .
To find the y-intercept, set x = 0 and solve for y : − 3 ( 0 ) + 11 y = 66 A rry = 6 . Thus, the y-intercept is ( 0 , 6 ) .
The x-intercept is ( − 22 , 0 ) .
The y-intercept is ( 0 , 6 ) . The final answer is: x-intercept ( − 22 , 0 ) , y-intercept ( 0 , 6 ) . x : ( − 22 , 0 ) , y : ( 0 , 6 )
Explanation
Understanding the Problem We are given the linear equation − 3 x + 11 y = 66 and we need to find the x and y intercepts. The x -intercept is the point where the line crosses the x -axis, which occurs when y = 0 . The y -intercept is the point where the line crosses the y -axis, which occurs when x = 0 .
Finding the x-intercept To find the x -intercept, we set y = 0 in the equation − 3 x + 11 y = 66 :
− 3 x + 11 ( 0 ) = 66 − 3 x = 66 Dividing both sides by − 3 , we get: x = − 3 66 = − 22 So, the x -intercept is ( − 22 , 0 ) .
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation − 3 x + 11 y = 66 :
− 3 ( 0 ) + 11 y = 66 11 y = 66 Dividing both sides by 11 , we get: y = 11 66 = 6 So, the y -intercept is ( 0 , 6 ) .
Final Answer Therefore, the x -intercept is ( − 22 , 0 ) and the y -intercept is ( 0 , 6 ) .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, if you have a cost function, the y-intercept represents the fixed costs (costs when production is zero), and the x-intercept could represent the break-even point (when costs equal revenue). Similarly, in physics, if you are analyzing motion, the intercepts on a graph of position versus time can tell you the starting position and when an object passes a certain point. These concepts provide a foundation for more advanced modeling and analysis in many fields.
