# 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

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

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

Average Rating     Jul 11, 2016 Rating     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 Rating     Math Problem by: Anonymous This helps a lot.thanks

 Feb 09, 2015 Rating     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 = xWrite an equation for Tracy based on the information given:3x-38 = 10Now solve Tracy's equation for x and this will tell you what Daniel spent.3x-38 +38 = 10+383x = 483x/3 = 48/3x = 16.Daniel spent \$16. Hope this helps.Karin

 Nov 26, 2014 Rating     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 = 16/24Bus = 7/24Motorcycle = 1/24Now we must write 2 equations so we must define 2 variables.Let y = total km for the tripLet 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 = x7/24y = x+3Solve the first equation for y.24(1/24)y = x*24 (Multiply by 24 on both sides)y = 24xSubstitute 24x for y into the 2nd equation.7/24(24x) = x+37x = x+37x - x = x-x +3 Subtract x from both sides6x = 36x/6 = 3/6 Divide by 6 on both sidesx = 1/2Now that we know x we can solve for y to answer the question.1/24y = x1/24y = 1/2(24/1)(1/24)y = 1/2(24/1) Multiply by 24/1 y = 12The total length of the trip was 12km.Check:Total = 12kmMotorcycle = .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
Rating     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