Travis has x nickels and y dimes. He has no more than 20 coins worth a minimum of $1.40 combined. Solve this system of inequalities graphically and determine one possible solution.

Question

Travis has xx nickels and yy dimes. He has no more than 20 coins worth a minimum of $1.40 combined. Solve this system of inequalities graphically and determine one possible solution.

Answer 1

The number of nickels (x) and dimes (y) is no more than 20:

x+y ≤ 20

Write this in slope-intercept form for ease in graphing:

y ≤ 20 - x

We can't have a negative number of either nickels

x ≥ 0

or dimes

y ≥ 0.

The total value of the coins is at least $1.40, so convert x and y to fractions of a dollar:

.05x + .1y ≥ 1.40

Write this in slope-intercept form for ease in graphing:

.1y ≥ 1.4 - .05x (multiply both sides by 100)

10y ≥ 140 - 5x (divide both sides by 10)

y ≥ 14 - (x/2)

Using Desmos to graph these constraints, we arrive at the following solution set (feasible region):

https://www.desmos.com/calculator/mpruudskao

One possible solutions is x=8 and y=12. Checking:

x+y = (8) + (12) ≤ 20

and

.05x + .1y = .05(8) + .1(12) = .4 + 1.2 = 1.6 = $1.60

So this is one solution. Because we can not have fractional nickels nor dimes, there several, but not infinitely many solutions.

Answer 2

Let

x represent the number of nickels and

y represent the number of dimes.

The number of coins is no more than 20:

x+y≤20

The total value of the coins is at least $1.40:

0.05x+0.10y≥1.40

Now, let's graph these inequalities on a coordinate plane.

Inequality 1:

x+y≤20

This is a linear inequality, and its graph is a line with a shaded region below the line.

Inequality 2:

0.05x+0.10y≥1.40

First, let's multiply both sides by 100 to eliminate decimals:

5x+10y≥140

Now, divide both sides by 5:

x+2y≥28

This is another linear inequality, and its graph is a line with a shaded region above the line.

Now, we can graph both inequalities on the same coordinate plane and find the region that satisfies both:

x+y≤20 ....1

x+2y≥28 ....2

On solving we got (12 ,8) on graph for (x,y)