Marcos is 7 years older than his sister, Rosa. The sum of their ages is 1 less than 3 times Rosa's age. How old is each?
m = r + 7
m + r + 1 = 3 r
so m = 2 r - 1
so
r+7 = 2 r - 1
8 = 1 r
Rosa is 8
m = 8 + 7 = 15
To solve this problem, we can set up a system of equations based on the given information.
Let's say Rosa's age is represented by "R", and Marcos' age is represented by "M".
1) Marcos is 7 years older than Rosa, so M = R + 7.
2) The sum of their ages is 1 less than 3 times Rosa's age, so M + R = 3R - 1.
Now we have a system of two equations:
Equation 1: M = R + 7
Equation 2: M + R = 3R - 1
We need to substitute the value of M from Equation 1 into Equation 2 to solve the system.
Substituting M = R + 7 into Equation 2, we get:
(R + 7) + R = 3R - 1
Now we can simplify this equation:
2R + 7 = 3R - 1
Subtracting 2R from both sides, we have:
7 = R - 1
Adding 1 to both sides, we get:
8 = R
Now that we know Rosa's age is 8, we can substitute this value into Equation 1 to find Marcos' age:
M = R + 7
M = 8 + 7
M = 15
Therefore, Rosa is 8 years old, and Marcos is 15 years old.