Both x and y are positive real numbers, and the point (x,y) lies on or above both of the lines having equations 2x + 5y = 10 and 3x + 4y = 12. What is the least possible value of 8x + 13y?
+128474 Look at the graph here : https://www.desmos.com/calculator/vsoqo96xtr
The least possible value of 8x + 13y will occur at the intersection of these lines
So we have that
2x + 5y =10
3x + 4y = 12 multiply the first eauation through by 4 and the second by -5
8x + 20y = 40
-15x - 20y = -60 add these
-7x = -20
x = 20/7 = the x coordinate of the intersection of these lines
And using 2x + 5y = 10 we can find the y coordinate as
2(20/7) + 5y =10
40/7 + 5y = 70/7
5y= 70/7 - 40/7
5y = 30/7 divide both sides by 5
y = 6/7
So.....the minimum value of 8x + 13y =
8(20/7) + 13(6/7) =
[160 + 78 ] / 7 =
238 / 7
34
+26367 Both x and y are positive real numbers,
and the point (x,y) lies on or above both of the lines having equations \(2x + 5y = 10\) and \(3x + 4y = 12\).
What is the least possible value of \(8x + 13y\)?
\(\begin{array}{|lrcll|} \hline & 2x + 5y &=& 10 \\ & 3x + 4y &=& 12 \\ & 3x + 4y &=& 12 \\ \hline \text{sum} & 8x + 13y &=& 10+12+12 \\ & \mathbf{8x + 13y} &=& \mathbf{34} \\ \hline \end{array}\)
