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Writing and Solving Systems of Equations

by Chuck

The sum of Eli's age and Cecil's age is 60. Six years ago, Eli was three times as old as Cecil.Find Eli's age now.



Karin from Algebra Class Says:



In order to solve this problem, you must write a system of equations.

The first equation is pretty easy to write.

The sum of Eli and Cecil's age is 60.

Let x = Eli's age
Let y = Cecil's age

Sum means add, so:
x + y = 60

The second equation is a little tricker.

Six years ago, Eli was three times as old as Cecil.

Six years ago means we have to subtract 6

Think:
Eli (six years ago) = 3 times cecil's age (six years ago)

x - 6 = 3(y-6)

Now, let's use the distributive property.

x - 6 = 3y - 18

Next we'll get the variables on one side and the constants on the other in order to write the equation in standard form.

x - 3y - 6 = 3y -3y - 18 Subtract 3y
x - 3y - 6 = -18

x - 3y - 6 + 6 = -18 + 6 Add 6

x - 3y = -12

Now we have two equation in standard form:

x + y = 60
x - 3y = -12

You can solve easily by using substitution or linear combinations.

I will use the linear combinations method.

Step 1: Create 1 set of opposite terms.

x + y = 60
-1 ( x - 3y) = -12(-1)



x + y = 60
-x + 3y = 12


Step 2: Add
x + y = 60
-x + 3y =12
--------------
4y = 72

Step 3: Solve for y

4y/4 = 72/4

y = 18

Cecil's age is 18

Step 4: Substitute to find Eli's age.

x + y = 60

x + 18 = 60
x+18 - 18 = 60 -18
x = 42

Eli's age is 42.

Check:
42 + 18 = 60

Six years ago:
Eli was 36
Cecil was 12

Eli was 3 times as old as Cecil 6 years ago.
12 * 3 = 36

Hope this helps,
Karin

Comments for Writing and Solving Systems of Equations

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Jul 11, 2016
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Please answer
by: Anonymous

Write a system of equations to model this problem and solve:

A shoe store costs 1800 dollars a month to operate. The average wholesale cost of each pair of shoes is 25 dollars, and the average price of each pair of shoes is 65 dollars. How many pairs of shoes must the store sell each month to break even?

Jun 02, 2016
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Math Problem
by: Anonymous

This helps a lot.thanks

Feb 09, 2015
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Math problem
by: Anonymous

While shopping for clothes, Tracy spent $38 less than 3 times what Daniel spent. Tracy spent $10. How much did Daniel spend?

____________________________________________________
Karin From Algebra Class Says:

Let Daniels' spending = x

Write an equation for Tracy based on the information given:
3x-38 = 10

Now solve Tracy's equation for x and this will tell you what Daniel spent.

3x-38 +38 = 10+38
3x = 48
3x/3 = 48/3
x = 16.

Daniel spent $16.

Hope this helps.
Karin

Nov 26, 2014
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Kindly solve this word equation
by: Anonymous

Jane travels 2/3 of her journey by train, 7/8 of the remaining journey by bus. The rest of the journey she rides a motorcycle. If the journey covered by bus is 3km longer than the journey covered by the motorcycle, How long is the journey in kms.

Kindly answer this question
__________________________________________________
Karin from Algebra Class Says:
This is a tricky question!

Step 1: You need to figure out the fraction of each part of the trip (with a common denominator among all three parts).
You know that the train is 2/3 of the entire trip.
The bus is 7/8 of the remaining part. So you have to think: 1/3 of the trip remains, so 7/8 of 1/3 means 7/8 *1/3 = 7/24.
If the train is 2/3, we must write that fraction with a common denominator of 24. 2/3 = 16/24.
So, train = 16/24 and bus = 7/24. This is a total of 23/24, so the motorcycle part of the trip was the 1/24 left over.
Train = 16/24
Bus = 7/24
Motorcycle = 1/24

Now we must write 2 equations so we must define 2 variables.
Let y = total km for the trip
Let x = total km for motorcycle journey.

(I picked x to be the motorcycle journey because the problem said: "If the journey covered by bus is 3km longer than the journey covered by the motorcycle)

Now we need to write 2 equations:
1/24y = x (1/24 of the total journey equals the motorcycle part of the trip)
7/24y = x+3 (7/24 of the total journey equals the motorcycle part plus 3 more km.)

Now that we have two equations, we can use the substitution method to solve:
1/24y = x
7/24y = x+3

Solve the first equation for y.
24(1/24)y = x*24 (Multiply by 24 on both sides)
y = 24x

Substitute 24x for y into the 2nd equation.
7/24(24x) = x+3
7x = x+3
7x - x = x-x +3 Subtract x from both sides
6x = 3
6x/6 = 3/6 Divide by 6 on both sides
x = 1/2

Now that we know x we can solve for y to answer the question.

1/24y = x
1/24y = 1/2
(24/1)(1/24)y = 1/2(24/1) Multiply by 24/1
y = 12

The total length of the trip was 12km.

Check:
Total = 12km
Motorcycle = .5km (This was our answer for x)
Bus = 3.5 (Bus is 3 more km than motorcycle)
Train = 8 km (12-4 = 8)

12(2/3) = 8 This verifies the train portion.
12(7/24) = 3.5 This verifies the bus portion.
12(1/24) = .5 This verifies the motorcycle portion.

Hope this helps.

Feb 22, 2012
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Please solve this
by: Anonymous

Please solve this:

a=4b---->E1
3a-2b=30----->E2


Karin from Algebra Class Says:




For this problem, you already have one equation solved for one variable: a = 4b

Step 1: Substitute 4b for a into the 2nd equation.

3(4b) - 2b =30
12b -2b =30
10b = 30
10b/10 = 30/10
b = 3

Step 2: Substitute 3 for b into the equation:
a = 4b
a = 4(3)
a = 12

Solution: a = 12 and b = 3

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