Find the [tex]$x$[/tex] and [tex]$y$[/tex] intercepts of the equation: [tex]$-5 x+7 y=140$[/tex]
The intercepts are:
[tex]$x$[/tex] intercept = ([tex]$\square$[/tex] ,0)
[tex]$y$[/tex] intercept = (0, [tex]$\square$[/tex])
Answer (2)
To find the x-intercept, set y = 0 and solve for x : − 5 x + 7 ( 0 ) = 140 , which gives x = − 28 .
To find the y-intercept, set x = 0 and solve for y : − 5 ( 0 ) + 7 y = 140 , which gives y = 20 .
The x-intercept is ( − 28 , 0 ) .
The y-intercept is ( 0 , 20 ) . The intercepts are ( − 28 , 0 ) and ( 0 , 20 ) .
Explanation
Understanding the Problem We are given the equation − 5 x + 7 y = 140 and asked to find the x and y intercepts. 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 substitute y = 0 into the equation: − 5 x + 7 ( 0 ) = 140 This simplifies to: − 5 x = 140 Dividing both sides by − 5 , we get: x = − 5 140 = − 28 So the x -intercept is ( − 28 , 0 ) .
Finding the y-intercept To find the y -intercept, we substitute x = 0 into the equation: − 5 ( 0 ) + 7 y = 140 This simplifies to: 7 y = 140 Dividing both sides by 7 , we get: y = 7 140 = 20 So the y -intercept is ( 0 , 20 ) .
Final Answer Therefore, the x -intercept is ( − 28 , 0 ) and the y -intercept is ( 0 , 20 ) .
Examples
Understanding intercepts is useful in many real-world scenarios. For example, if you are analyzing a budget equation where x represents the number of one item purchased and y represents the number of another item, the intercepts tell you the maximum quantity of each item you can buy if you spend all your money on just that item. Similarly, in physics, if you have an equation describing the motion of an object, the intercepts can represent the starting position or the time when the object reaches a certain point.
The x -intercept of the equation − 5 x + 7 y = 140 is ( − 28 , 0 ) , and the y -intercept is ( 0 , 20 ) . These points represent where the line crosses the x and y axes respectively.
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