Mia has a certain amount of money. If she buys 4 pens and 1 pencil, she will have $5 left over. If she buys 2 pens and 2 pencils, she will have $3 left over. If Edward arrives with the same amount of money as Mia, together they can buy 7 pens and 4 pencils and spend all their money. If \(a\) is the cost of one pen and \(b \) is the cost of one pencil, compute the ordered pair \((a,b) \).
Note: This is a long and detailed answer!
Let the amount of money that Mia has =M=same amount of money as Edward has
M - [4a + b] =5
M - [2a + 2b]=3
[7a + 4b] =2M
Solve the following system:
{-4 a - b + M = 5 | (equation 1)
-2 a - 2 b + M = 3 | (equation 2)
7 a + 4 b = 2 M | (equation 3)
Express the system in standard form:
{-(4 a) - b + M = 5 | (equation 1)
-(2 a) - 2 b + M = 3 | (equation 2)
7 a + 4 b - 2 M = 0 | (equation 3)
Swap equation 1 with equation 3:
{7 a + 4 b - 2 M = 0 | (equation 1)
-(2 a) - 2 b + M = 3 | (equation 2)
-(4 a) - b + M = 5 | (equation 3)
Add 2/7 × (equation 1) to equation 2:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a - (6 b)/7 + (3 M)/7 = 3 | (equation 2)
-(4 a) - b + M = 5 | (equation 3)
Multiply equation 2 by 7/3:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a - 2 b + M = 7 | (equation 2)
-(4 a) - b + M = 5 | (equation 3)
Add 4/7 × (equation 1) to equation 3:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a - 2 b + M = 7 | (equation 2)
0 a+(9 b)/7 - M/7 = 5 | (equation 3)
Multiply equation 3 by 7:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a - 2 b + M = 7 | (equation 2)
0 a+9 b - M = 35 | (equation 3)
Swap equation 2 with equation 3:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a+9 b - M = 35 | (equation 2)
0 a - 2 b + M = 7 | (equation 3)
Add 2/9 × (equation 2) to equation 3:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a+9 b - M = 35 | (equation 2)
0 a+0 b+(7 M)/9 = 133/9 | (equation 3)
Multiply equation 3 by 9/7:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a+9 b - M = 35 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
Add equation 3 to equation 2:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a+9 b+0 M = 54 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
Divide equation 2 by 9:
{7 a + 4 b - 2 M = 0 | (equation 1)
0 a+b+0 M = 6 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
Subtract 4 × (equation 2) from equation 1:
{7 a + 0 b - 2 M = -24 | (equation 1)
0 a+b+0 M = 6 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
Add 2 × (equation 3) to equation 1:
{7 a+0 b+0 M = 14 | (equation 1)
0 a+b+0 M = 6 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
Divide equation 1 by 7:
{a+0 b+0 M = 2 | (equation 1)
0 a+b+0 M = 6 | (equation 2)
0 a+0 b+M = 19 | (equation 3)
a = $2, b=$6, M=$19
M - [4a + b] =5.............(1)
M - [2a + 2b]=3...........(2)
[7a + 4b] =2M.............(3)
M=5 + 4a + b -rearrange(1).....................(4)
M=3 + 2a + 2b -rearrange(2) and subt.....(5)
0 =2 + 2a - b.............................................(6)
2M=0 + 7a + 4b - rearrange (3)
2M=10 + 8a + 2b - Multiply (4) by 2 and subt.
0 =-10 - a + 2b.........................................(7)
Solve (6) and (7) as 2 simult.equations.
a=$2 and b=$6. Sub these into (1)
M =$19
