Question

How do you graph the solution set of `(4x-5y)<=20` and `2x+2y>=4`?

Answer 1

First, solve each equation for y, putting it in standard form.
Next, graph the lines and lightly shade the area that fulfills the inequality. Then, darkly shade the area that has overlapping light shading. This area is the solution of the system of inequalities.

Explanation:

For graphing online, I would recommend https://www.desmos.com/calculator as a user-friendly and free online graphing calculator.

I graphed the two equations here.

How to find it algebraically is below:
First, solve each equation for y, putting it in standard form:
`4x-5y<=20`
`color(red)(y >= 4/5x-4)` The sign flipped because we divided by a negative number.
`2x+2y>=4`
`color(green)(y >= x + 2)`

Next, graph the lines and lightly shade the area that fulfills the inequality. Then, darkly shade the area that has overlapping light shading. This area is the solution of the system of inequalities.

Bonus: Finding the point of intersection. This point is where to two lines intersect and is also the minimum of this system.

First, set it up as a system of equations and change the inequality symbols to equals signs:
`4x-5y = 20`
`2x+2y = 4`

Now solve for one variable using either substitution or subtraction. I will use subtraction:
`4x-5y=20`
`4x+4y=8`
-_____
`0x-9y=12`
`y = -4/3`

Then, plug your variable into the original equation:
`4x-5(-4/3) = 20`
`4x+(20/3) = 20`
`12x+20 = 60`
`12x = 40`
`x = 10/3`

The point of intersection is `(10/3, -4/3)`