# Question Video: Finding the Point That Maximizes the Objective Function given the Graph of the Constrains Mathematics • 9th Grade

Determine the values of π₯ and π¦ that maximize the function π = 5π₯ + 2π¦. Write your answer as a point (π₯, π¦).

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### Video Transcript

Determine the values of π₯ and π¦ that maximise the function π equals five π₯ plus two π¦. Write your answer as a point π₯, π¦.

Weβve been given a region which represents a series of inequalities. The topmost horizontal inequality is π¦ is less than or equal to eight. The vertical inequality is π₯ is less than or equal to seven. The region is also bound by the π₯-axis. So π¦ must be greater than or equal to zero. And finally, we have a diagonal line with a π¦-intercept at eight and a gradient of negative eight over three.

This inequality is, therefore, π¦ is greater than or equal to negative eight over three π₯ plus eight. All maxima and minima occur at the corners of the region. Weβre specifically interested in the maximum of the function. So by finding the value of π₯ and π¦ at these points, we can determine which order pair when substituted into π equals five π₯ plus two π¦ will give us the largest possible value for π.

Our vertices fall at seven, zero; seven, eight; zero, eight; and three, zero. At zero, eight, π is equal to five times zero, add two times eight, which gives us a value of 16. At three, zero, π is equal to five times three, add two times zero, which is 15. At seven, eight, π is equal to five times seven, add two times eight, which is 51. And at seven, zero, π is equal to five times seven, add two times zero, which is 35.

Our function π equals five π₯ plus two π¦ is therefore greatest when π₯ equals seven and π¦ equals eight. The values of π₯ and π¦ that maximise our function are seven, eight.