Solving Systems of Linear Equations using Substitution

a

Let $b$ be the number of backpacks with tent and $w$ be the number of warm hiking clothes they rent. Since they in total rent $16$ items, we can write the first equation as: $b + w = 16 .$ We also know that the hikers spend $$ 256$ to rent equipment from Norgay High Altitute Hiking. Additionally, a backpack+tent, $b,$ costs $$ 25$ and a set of warm hiking clothes, $w,$ costs $$ 13 .$ We can write this as the equation: $25 b + 13 w = 256 .$ Combining the equations, we have the following system of equations: ${b + w = 16 25 b + 13 w = 256 .$

b

To find how many backpack+tents and how many sets of warm hiking clothes the group of hikers rent, we have to solve the system

{b + w = 16 25 b + 13 w = 256 (I) (II)

for

b

and

w .

We will use the substitution method. In order to use it, we need one of the variables isolated. Let's isolate

b

in Equation (I).

b + w = 16 \Leftrightarrow b = 16 - w

Now we can substitute

16 - w

for

b

in Equation (II) and solve the resulting equation for

w .

25 b + 13 w = 256

Substitute

b = 16 - w

25 (16 - w) + 13 w = 256

DistrDistribute

25

400 - 25 w + 13 w = 256

AddTermsAdd terms

400 - 12 w = 256

SubEqn

LHS - 400 = RHS - 400

- 12 w = - 144

DivEqn

LHS / - 12 = RHS / - 12

w = 12

Now we can find

b

by substituting

12

for

w

in Equation (I) and solve the resulting equation for

b .

b + w = 16

Substitute

w = 12

b + 12 = 16

SubEqn

LHS - 12 = RHS - 12

b = 4

Thus, the hikers rents

4

backpacks+tents and

12

sets of warm hiking clothes.

Solving Systems of Linear Equations using Substitution 1.8 - Solution