A line has the general linear equation $7x - 4y = 28$.
(a) 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 ( 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 ( 0 , − 7 ) . The x -intercept is ( 4 , 0 ) and the y -intercept is ( 0 , − 7 ) .
Explanation
Understanding the Problem We are given the equation of a line 7 x − 4 y = 28 , and we need to find its x - and y -intercepts. The x -intercept is the point where the line crosses the x -axis, meaning y = 0 . The y -intercept is the point where the line crosses the y -axis, meaning x = 0 .
Finding the x-intercept To find the x -intercept, we substitute y = 0 into the equation 7 x − 4 y = 28 . This gives us 7 x − 4 ( 0 ) = 28 , which simplifies to 7 x = 28 . Dividing both sides by 7, we get x = 7 28 = 4 . Therefore, the x -intercept is ( 4 , 0 ) .
Finding the y-intercept To find the y -intercept, we substitute x = 0 into the equation 7 x − 4 y = 28 . This gives us 7 ( 0 ) − 4 y = 28 , which simplifies to − 4 y = 28 . Dividing both sides by -4, we get y = − 4 28 = − 7 . Therefore, the y -intercept is ( 0 , − 7 ) .
Final Answer 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 can represent the break-even point, where costs are covered. Similarly, in physics, intercepts on a graph of motion can indicate starting points or points of rest. Knowing how to find intercepts helps in interpreting data and making informed decisions in these fields.
