Question

three bags of potatoes and four cases of corn cost $40. Five bags of potatoes and two cases of corn cost $34. Find the cost of one bag of potatoes and the cost of one case of corn.

Answer 1

Use variables and set up equations.

$3p + 4c = 40 \\ 5p + 2c = 34$

Next, get one variable with equal coefficients.

$3p + 4c = 40 \\ 10p + 4c = 68\ (multiplied\ both\ sides\ by\ 2)$

Then get it alone on one side.

$4c = 40-3p \\ 4c = 48-10p$

Now we can use substitution. Since $40-3p = 4c$, substitute $40-3p$ for $4c$ in the other equation and you'll get an equation with just one variable to simplify and solve.

$40-3p = 68-10p \\ 40+7p = 68 \\ 7p = 28 \\ \boxed{p = 4}$

We can then use $p = 4$ in an earlier equation to find $c$.

$5p + 2c = 34 \\ 5*4 + 2c = 34 \\ 20 + 2c = 34 \\ 2c = 14 \\ \boxed{c=7}$

Answer 2

3p + 4c = 40 ⇒ 15p + 20c = 200
5p + 2c = 34 ⇒ 15p +   6c = 102
                                    14c = 98
                                     14     14
                                        c = 7
                            5p + 2(7) = 34
                              5p + 14 = 34
                                    - 14  - 14
                                      5p = 20
                                       5      5
                                        p = 4
                                  (p, c) = (4, 7)