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Systems of Equations Problems

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A florist buys 302 flowers to make corsages for the Winter Dance. A carnation corsage used three flowers and a rose corsage uses two flowers. If the florist made 112 corsages, how many of each type were made?

Let c = number of carnation corsages and r = number of rose corsages.
NUMBER OF CORSAGES: c + r = 112
FLOWERS USED: 3c + 2r = 302
Multiply top equation by -2:
-2c - 2r = -224
3c + 2r = 302
c = 78
r = 34
Answer: 78 carnation corsages and 34 rose corsages.

In a two-digit number, the tens digit is 4 more than the units digit. The number is 7 less than 8 times the sum of the digits. Find the number.

t = u + 4
"The number" is "7 less than 8 times the sum of the digits."
10t + u = 8(t + u) - 7
Substitute t = u + 4
10(u + 4) + u = 8(u + 4 + u) - 7
10u + 40 + u = 8u + 32 + 8u - 7
11u + 40 = 16u + 25
15 = 5u
u = 3
t = 7
The number is 73

The sum of the digits of a two-digit number is 12. If the digits are reversed, the number is decreased by 18. What is the original number?

t + u = 12
Reversed number is 18 less than the original.
10u + t = 10t + u - 18    Subtract u and t from each side.
9u = 9t - 18    Divide by 9
u = t - 2
Substitute into original equation.
t + (t - 2) = 12
2t = 14
t = 7
u = 5
ANS: Original number is 75.

The bill for 3 glasses of orange juice and 5 pancake specials is $7.60, but the bill for 5 glasses of orange juice and 4 pancake specials is $7.90. What would be the bill for one glass of orange juice and one pancake special?

3j + 5p = 7.60
5j + 4p = 7.90
Multiply top equation by 5 and bottom one by -3.
15j + 25p = 38
-15j - 12p = -23.70 Add these two equations.
13p = 14.30
p = 1.10
Substitute into first equation.
3j + 5(1.10) = 7.60
3j + 5.50 = 7.60
3j = 2.10
j = .70
ANS: 1 glass of juice + 1 pancake special = $.70 + $1.10 = $1.80

The method used to solve a system of 2 equations in 2 variables can be extended to solve a system of 3 equations in 3 variables. Follow the steps below to solve the following system:
x + y + z = 2
x - y + z = -2
x + y - 2z = 5

1. Label the equations #1, #2, #3.
2. Work with equations #1 and #2 to eliminate one of the variables.
3. Label the resulting equation #4.
4. Work with equations #2 and #3 to eliminate the SAME variable you eliminated in step 2.
5. Label the resulting equation #5.
6. Work with equations #4 and #5 just as you would to solve any 2-variable system.
7. Substitute the results from step 6 into any of the original equations to find the last variable.
8. Check your answer in the original equations!

Add equations 1 and 2: EQN 4: 2x + 2z = 0
Add equations 2 and 3: EQN 5: 2x - z = 3
Subtract equations 4 and 5: 3z = -3
z = -1
Substitute into EQN 4: 2x - 2 = 0
x = 1
Substitute into EQN 1: 1 + y - 1 = 2
y = 2

Solve this system of equations by substitution:

• y = 2x² + 2x + 3
• x = y - 3

Substitute y - 3 for x in the top equation:
y = 2(y - 3)² + 2(y - 3) + 3
y = 2(y² - 6y + 9) + 2y - 6 + 3
y = 2y² - 12 y + 18 + 2y - 6 + 3
y = y² - 10y + 15
0 = y² - 11y + 15
0 = (2y - 5)(y - 3)
2y - 5 = 0 OR y - 3 = 0
y = 2.5 OR y = 3

If y = 2.5, x = -.5 ---> (-.5, 2,5)
If y = 3, x = 0 ---> (0, 3)

Try checking these results with a graphing calculator or EXCEL!

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