Four years from now Emily will be as old as Cathy is now. Seven years ago, the sum of their ages was 22. Write a system of equations to model this situation and solve it using substitution to determine how old the girls are now.

Question

Four years from now Emily will be as old as Cathy is now. Seven years ago, the sum of their ages was 22. Write a system of equations to model this situation and solve it using substitution to determine how old the girls are now.

Answer 1

e+4=c

e-7 +c-7 = 22

e - c = -4
e + c = 36
-------------- add
2 e = 32
so
e = 16
then c = 20

Answer 2

Let's use the following variables to represent the ages of the girls:

- E for Emily's age now
- C for Cathy's age now

We are given that four years from now, Emily's age will be the same as Cathy's age now. This can be represented as:

E + 4 = C

We are also given that the sum of their ages seven years ago was 22:

(E - 7) + (C - 7) = 22

To solve this system of equations using substitution, we can rearrange the first equation to solve for C in terms of E:

C = E + 4

Substituting this expression for C in the second equation, we have:

(E - 7) + (E + 4 - 7) = 22

Simplifying, we get:

2E - 10 = 22

Adding 10 to both sides, we have:

2E = 32

Dividing both sides by 2, we find:

E = 16

Substituting this value back into the first equation to solve for C, we have:

C = 16 + 4 = 20

Therefore, Emily is currently 16 years old and Cathy is currently 20 years old.

Answer 3

Let's let x represent Emily's current age, and y represent Cathy's current age.

From the given information, we can create two equations:

1) "Four years from now Emily will be as old as Cathy is now."
This can be translated to:
x + 4 = y

2) "Seven years ago, the sum of their ages was 22."
This can be translated to:
(x - 7) + (y - 7) = 22

Now we have a system of two equations:

1) x + 4 = y
2) (x - 7) + (y - 7) = 22

To solve this system using substitution, we will solve equation 1 for y and substitute it into equation 2.

From equation 1, we get: y = x + 4

Now we substitute (x + 4) for y in equation 2:

(x - 7) + ((x + 4) - 7) = 22

Simplifying:

x - 7 + x + 4 - 7 = 22
2x - 10 = 22

Next, we solve for x:

2x - 10 = 22
2x = 32
x = 16

Now we substitute x = 16 back into equation 1 to find y:

y = x + 4
y = 16 + 4
y = 20

So Emily is currently 16 years old and Cathy is currently 20 years old.