# Linear Programming

How to solve linear programming problems: example and its solution.

## Example

Make a table to write

two linear inequalities and one linear equation.

Set the number of chocolate milk as *x*

and the number of a chocolate shake as *y*.

Write 'Choco Milk (*x*)' and 'Chocolate Shake (*y*)'

as the titles of 2nd and 3rd column.

Fill each row using each condition.

Powder:

1⋅*x* spoons are needed

to make *x* cups of choco milk.

2⋅*y* spoons are needed

to make *y* cups of choco shake.

There are 10 spoons of powder.

So 1⋅*x* + 2⋅*y* ≤ 10.

Milk:

2⋅*x* cups are needed

to make *x* cups of choco milk.

1⋅*y* cups are needed

to make *y* cups of choco shake.

There are 11 cups of milk.

So 2⋅*x* + 1⋅*y* ≤ 11.

Sales:

1⋅*x* dollars income from *x* cups of choco milk.

1.5⋅*y* dollars income from *y* cups of choco shake.

Let's say the total sales is $*k*.

Then 1⋅*x* + 1.5⋅*y* = *k*

Write the powder inequality in slope-intercept form.*y* ≤ -(1/2)*x* + 5

It's for graphing and comparing the slopes.

Write the milk inequality in slope-intercept form.*y* ≤ -2*x* + 11

Write the sales equation in slope-intercept form.*y* ≤ (-2/3)*x* + 2/3*k*

Draw the inequalities on the coordinate plane.

Color the intersecting region. (blue region)

(Of course *x* ≥ 0, *y* ≥ 0.)

System of linear inequalities

Then think about the sales equation:*y* ≤ (-2/3)*x* + 2/3*k*. (red line)

To maximize *k* (sales),

the *y*-intercept, + 2/3*k*, has to be maximized.

And the slope is (-2/3),

which is between -1/2 and -2.

Then, to maximize + 2/3*k*,

the red line has to pass the red point:

the intersecting point of

the 'powder equation' and the 'milk equation'.

To find the red point,

solve the system of

the powder equation (*x* + 2*y* = 10)

and the milk equation (2*x* + *y* = 11).

Then the red point is (4, 3).

System of linear equations: elimination method

So the sales (*k*) is maximized

if the sales equation (*x* + 1.5*y* = *k*)

passes through (4, 3).

So put (4, 3) into the sales equation.

Then *k* = 8.5.

This means the maximum sales is $8.5.