Question

When solving systems of equations we have at least two unknowns. A common example of a system of equations is a price problem. For example, Jacob has 60 coins consisting of quarters and dimes. The coins combined value is $9.45. Find out how many of each (quarters and dimes) Jacob has. What do the unknowns in this system represent and what are the two equations that that need to be solved? Finally, solve the system of equations.

Answer 1

Answer:

a. The unknowns in the equation above are q and d that represent the number of quarters and dimes respectively.

b. The two equations that need to be solved are; q+d=60...equation 1, and

0.25 q+0.1 d=9.45...equation 2

c.

Number of dimes, d=37

Number of quarters, q=23

Explanation:

a.

Step 1: Derive equation for determining total number of coins.

The total number of coins can be derived using the expression below;

T=q+d

where;

T=total number of coins

q=number of quarters

d=number of dimes

In our case;

T=60 coins

q=unknown

d=unknown

replacing;

q+d=60...equation 1

The unknowns in the equation above are q and d that represent the number of quarters and dimes respectively.

b.

Step 2: Derive equation for determining combined value of coins.

V=(q×Q)+(d×D)

where;

V=total value of the coins

q=number of quarters

Q=unit value of a quarter in dollars

d=number of dimes

D=unit value of a dime in dollars

In our case;

V=$9.45

q=unknown

Q=$0.25

d=unknown

D=$0.10

replacing;

9.45=(0.25×q)+(0.10×d)

9.45=0.25 q+0.1 d

0.25 q+0.1 d=9.45...equation 2

Step 3: Combine equation 1 and 2 and solve simultaneously

(q+d=60)×0.25 >>>>>>>>>>>> 0.25 q+0.25 d=15

                                                 -

(0.25 q+0.1 d=9.45)×1>>>>>>> 0.25 q+0.1 d=9.45

(0.25 q-0.25 q)+(0.25 d-0.1 d)=15-9.45

0 q+0.15 d=5.55

0.15 d=5.55

d=5.55/0.15=37

d=37

Substitute the value of d in equation 1;

q+d=60

and q=37

37+q=60

q=60-37=23

q=23