Question

Andrea is buying some new shirts and sweaters. She is able to buy 6 shirts and 7 sweaters for $212 or she is able to buy 10 shirts and 8 sweaters for $258. How much does a shirt cost? How much does a sweater cost?

Answer 1

Sweater: $26

Shirt: $5

Explanation:

Systems of Equations:

We can set up two linear equations to form a systems of equations which will allow us to solve for the price of a shirt and sweater.

The first step we want to take is assigning the shirt and sweater to an unknown, so I'll just refer to the price of a shirt as:
`S_h` with the little subscript "h" and I'll refer to the price of a sweater as:
`S_w` with the little subscript "w".

Now the next main thing to understand is we can calculate the total price of something by multiplying the price of an individual item by the quantity. So if I buy an apple for $10, but I buy 10 of them, then the total price would be 10 * 10 or $100. So when Andrea buys 6 shirts, the price of that can be represented as:
`6S_h` and when Anrea buys 7 sweaters the price of that can be represented as:
`7S_w`, adding these together should equal 212, since it gives us that buying 6 shirts and 7 sweaters costs a total of $212.

Using this logic with the other case where she buys 10 shirts and 8 sweaters for $258, we can set up the two equations:

`6S_h + 7S_w=212\\10S_h+8S_w=258`

From here we can use substitution to solve this equation. So let's solve for the
`S_h` on the top equation.

`6S_h+7S_w=212`

Froom here subtract
`7S_w` from both sides and then divide by 6

`S_h=(212-7S_w)/(6)`

Now we can plug this into the second equation:

`10((212-7S_w)/(6))+8S_2=258`

Simplifying the division and multiplication on the left side we get:

`(1060)/(3)-(70)/(6)S_w+8S_w=258`

From here we can combine the like terms

`(1060)/(3)-(22)/(6)S_w=258`

we can simplify the fraction a bit more

`(1060)/(3)-(11)/(3)S_w=258`

Now subtract 1060/3 from both sides

`(-11)/(3)S_w=(-286)/(3)`

Now divide both sides by -11/3

`S_w=26`

Now from here we can substitute this back into any equation to solve for the shirt price. Let's use the equation where we solved for the price of the shirt but we just needed to know the price of the sweater.

`S_h=(212-7S_w)/(6)`

Now let's plug in 26 for the sweater price.

`S_h=(212-7(26))/(6)`

Simplify this and we get:

`S_h=5`

and now we know the shirt and sweater price!