Find the $x$ and $y$ intercepts of this linear equation.
$-3 x+4 y=12$
x-intercept (____, ____)
y-intercept (____, ____)
Answer (1)
To find the x -intercept, substitute y = 0 into the equation and solve for x , resulting in x = − 4 .
To find the y -intercept, substitute x = 0 into the equation and solve for y , resulting in y = 3 .
The x -intercept is ( − 4 , 0 ) .
The y -intercept is ( 0 , 3 ) .
The x-intercept is − 4 , the y-intercept is 3 .
Explanation
Understanding the Problem We are given the linear equation − 3 x + 4 y = 12 and asked 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 substitute y = 0 into the equation and solve for x :
− 3 x + 4 ( 0 ) = 12 − 3 x = 12 x = − 3 12 x = − 4 So the x -intercept is ( − 4 , 0 ) .
Finding the y-intercept To find the y -intercept, we substitute x = 0 into the equation and solve for y :
− 3 ( 0 ) + 4 y = 12 4 y = 12 y = 4 12 y = 3 So the y -intercept is ( 0 , 3 ) .
Final Answer Therefore, the x -intercept is ( − 4 , 0 ) and the y -intercept is ( 0 , 3 ) .
Examples
Understanding intercepts is crucial in various real-world scenarios. For instance, in economics, the x-intercept of a cost function could represent the break-even point, where costs equal revenue. Similarly, in physics, the y-intercept of a velocity-time graph could represent the initial velocity of an object. By finding intercepts, we can gain valuable insights into the behavior of linear relationships in diverse fields.
