A line has the general form $7x - 4y = 28$. Find the $x$- and $y$-intercepts of the graph of this line.
$x$-intercept $(x, y) = ?$
$y$-intercept $(x, y) = ?$
Answer (1)
To find the x -intercept, set y = 0 in the equation and solve for x , resulting in x = 4 .
The x -intercept is therefore ( 4 , 0 ) .
To find the y -intercept, set x = 0 in the equation and solve for y , resulting in y = − 7 .
The y -intercept is therefore ( 0 , − 7 ) . The intercepts are ( 4 , 0 ) ; ( 0 , − 7 )
Explanation
Understanding the Problem We are given the equation of a line as 7 x − 4 y = 28 . Our goal is to find the x - and y -intercepts of this line. 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 7 x − 4 y = 28 :
7 x − 4 ( 0 ) = 28 7 x = 28 Dividing both sides by 7, we get: x = 7 28 = 4 So, the x -intercept is the point ( 4 , 0 ) .
Finding the y-intercept To find the y -intercept, we substitute x = 0 into the equation 7 x − 4 y = 28 :
7 ( 0 ) − 4 y = 28 − 4 y = 28 Dividing both sides by -4, we get: y = − 4 28 = − 7 So, the y -intercept is the point ( 0 , − 7 ) .
Final Answer Therefore, the x -intercept is ( 4 , 0 ) and the y -intercept is ( 0 , − 7 ) .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, the x-intercept of a cost function could represent the break-even point, while the y-intercept represents the fixed costs. Similarly, in physics, if you're analyzing the motion of an object, the intercepts on a graph of position vs. time can tell you the starting point and when the object reaches a certain location. Knowing how to find these intercepts helps in making informed decisions and predictions in these scenarios.
