Find the $x$ and $y$ intercepts of this linear equation.
$4 x+7 y=28$
x-intercept (?, ?)
y-intercept (?, ?)
Answer (1)
To find the x -intercept, set y = 0 and solve for x : 4 x + 7 ( 0 ) = 28 , which gives x = 7 . The x -intercept is ( 7 , 0 ) .
To find the y -intercept, set x = 0 and solve for y : 4 ( 0 ) + 7 y = 28 , which gives y = 4 . The y -intercept is ( 0 , 4 ) .
The x -intercept is ( 7 , 0 ) .
The y -intercept is ( 0 , 4 ) . Therefore, the final answer is: x -intercept ( 7 , 0 ) and y -intercept ( 0 , 4 ) .
Explanation
Understanding the Problem We are given the equation of a line: 4 x + 7 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 set y = 0 in the equation 4 x + 7 y = 28 . This gives us: 4 x + 7 ( 0 ) = 28
4 x = 28
Dividing both sides by 4, we get: x = 4 28 = 7
So, the x -intercept is the point ( 7 , 0 ) .
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation 4 x + 7 y = 28 . This gives us: 4 ( 0 ) + 7 y = 28
7 y = 28
Dividing both sides by 7, we get: y = 7 28 = 4
So, the y -intercept is the point ( 0 , 4 ) .
Final Answer Therefore, the x -intercept is ( 7 , 0 ) and the y -intercept is ( 0 , 4 ) .
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 that good alone. 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 linear relationships in many fields.
