Find the $x$-intercept and $y$-intercept of the line.
$6 x-4 y=24$
Answer (2)
To find the x -intercept, set y = 0 in the equation and solve for x : 6 x − 4 ( 0 ) = 24 , which gives x = 4 .
To find the y -intercept, set x = 0 in the equation and solve for y : 6 ( 0 ) − 4 y = 24 , which gives y = − 6 .
The x -intercept is 4 , meaning the line crosses the x -axis at the point ( 4 , 0 ) .
The y -intercept is − 6 , meaning the line crosses the y -axis at the point ( 0 , − 6 ) .
The x -intercept is 4 and the y -intercept is − 6 .
Explanation
Understanding the Problem We are given the equation of a line: 6 x − 4 y = 24 . Our goal is to find the x -intercept and the y -intercept of this line. 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 set y = 0 in the equation 6 x − 4 y = 24 . This gives us: 6 x − 4 ( 0 ) = 24
6 x = 24
Calculating the x-intercept Now, we solve for x :
x = 6 24 = 4
So, the x -intercept is at the point ( 4 , 0 ) .
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation 6 x − 4 y = 24 . This gives us: 6 ( 0 ) − 4 y = 24
− 4 y = 24
Calculating the y-intercept Now, we solve for y :
y = − 4 24 = − 6
So, the y -intercept is at the point ( 0 , − 6 ) .
Final Answer Therefore, the x -intercept is 4 and the y -intercept is − 6 .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, if the equation represents a budget constraint, the intercepts show the maximum amount of each good you can buy if you spend all your money on it. Similarly, in physics, if the equation represents the motion of an object, the intercepts can represent the initial position or the time when the object reaches a certain point. Knowing how to find intercepts helps in interpreting and analyzing such scenarios effectively.
The x -intercept of the line 6 x − 4 y = 24 is 4, meaning it crosses the x -axis at the point ( 4 , 0 ) . The y -intercept is -6, meaning it crosses the y -axis at the point ( 0 , − 6 ) . Thus, the x -intercept is 4 and the y -intercept is -6.
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