We can use the given information to create a system of equations. Then, solving the system for the variables will tell us how many small and large prints the artist will need to sell in order to break even. Let $x$ represent the number of small prints sold and $y$ represent the number of large prints sold.

Writing a System

We are told that the artist will sell small prints for

$ 20

each and large prints for

$ 45

each. It is also given that he needs to earn a total of

$ 510 .

Using the information, we can write the equation.

20 x + 45 y = 510

It is also given that the artist wishes to sell

twice

as many small prints. This gives us the second equation.

x = 2 y

Now, let's write the complete system of equations.

{20 x + 45 y = 510 x = 2 y (I) (II)

Solving a System

We will solve this system for the values of

x

and

y

using the Substitution Method. Specifically, we will substitute the second equation into the first equation and solve for

y .

{20 x + 45 y = 510 x = 2 y (I) (II)

Substitute

$(I):$ $x = 2 y$

{20 (2 y) + 45 y = 510 x = 2 y

Multiply

$(I):$ Multiply

{40 y + 45 y = 510 x = 2 y

AddTerms

$(I):$ Add terms

{85 y = 510 x = 2 y

DivEqn

$(I):$ $LHS / 85 = RHS / 85$

{y = 6 x = 2 y

Now that we have determined that

y = 6,

we can substitute this into the second equation.

{y = 6 x = 2 y (I) (II)

Substitute

$(II):$ $y = 6$

{y = 6 x = 2 (6)

Multiply

$(II):$ Multiply

{y = 6 x = 12

Since

x = 12

and

y = 6,

the artist will need to sell

12

small prints and

6

large prints to break even.

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