Let

x

be the number of registrations and

y

be the number of T-shirts bought each year by the student council. This year the registration for a conference costs

$ 15,

and the T-shirt costs

$ 10 .

By multiplying

15

by

x

and

10

by

y

and adding these expressions, we can find how much the student council will spend for the conference.

15 x + 10 y

We are also told that this sum is

$ 180 .

Let's set

180

equal to the above expression and form an equation.

15 x + 10 y = 180

Last year, they spent

$ 12

per registration and

$ 9

per T-shirt for a total of

$ 150

to buy the same number of registrations and T-shirts. Similarly, multiplying

12

by

x

and

9

by

y

and adding these expressions, we can find how much they spent in total. Then, we can set it equal to

$ 150

to form an equation.

12 x + 9 y = 150

Using the first and second equation, we can write the following system of equations.

{15 x + 10 y = 180 12 x + 9 y = 150 (I) (II)

Let's solve it by substitution. We will solve Equation (I) for

y

and substitute its equivalent value into Equation (II).

{15 x + 10 y = 180 12 x + 9 y = 150 (I) (II)

▼

(I):

Solve for

y

SubEqn

$(I):$ $LHS - 15 x = RHS - 15 x$

{10 y = 180 - 15 x 12 x + 9 y = 150

DivEqn

$(I):$ $LHS / 10 = RHS / 10$

{y = 18 - 1.5 x 12 x + 9 y = 150

Substitute

$(II):$ $y = 18 - 1.5 x$

{y = 18 - 1.5 x 12 x + 9 (18 - 1.5 x) = 150

▼

(II):

Solve for

x

Distr

$(II):$ Distribute $9$

{y = 18 - 1.5 x 12 x + 162 - 13.5 x = 150

SubTerm

$(II):$ Subtract term

{y = 18 - 1.5 x - 1.5 x + 162 = 150

SubEqn

$(II):$ $LHS - 162 = RHS - 162$

{y = 18 - 1.5 x - 1.5 x = - 12

DivEqn

$(II):$ $LHS / (- 1.5) = RHS / (- 1.5)$

{y = 18 - 1.5 x x = 8

Substitute

$(I):$ $x = 8$

{y = 18 - 1.5 (8) x = 8

▼

(I):

Solve for

y

Multiply

$(I):$ Multiply

{y = 18 - 12 x = 8

SubTerm

$(II):$ Subtract term

{y = 6 x = 8

The student council bought

8

registrations for the conference and

6

T-shirts each year.

Exercise

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