Answer:
The elder brother is 13 years old
Step-by-step explanation:
System of equations
Let's set:
x=current age of the younger brother
y=current age of the elder brother
The first condition states the sum of their ages is 22:
x + y = 22
It follows that:
x = 22 - y
Their ages six years ago were: x-6 and y-6. The product of both is 21:
( x - 6 ) ( y - 6 ) = 21
Replacing the expression of x:
( 22 - y - 6 ) ( y - 6 ) = 21
Simplifying:
(16 - y ) ( y - 6 ) = 21
Multiplying:
[tex]16y - 96 - y^2+6y=21[/tex]
Rearranging and simplifying:
[tex]y^2-22y+117=0[/tex]
Applying the quadratic solver:
[tex]\displaystyle y=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Where a=1, b=-22, c=117
[tex]\displaystyle y=\frac{22\pm \sqrt{(-22)^2-4(1)(117)}}{2(1)}[/tex]
[tex]\displaystyle y=\frac{22\pm \sqrt{16}}{2}[/tex]
[tex]\displaystyle y=\frac{22\pm 4}{2}[/tex]
There are two possible solutions:
[tex]\displaystyle y=\frac{22+ 4}{2}=13[/tex]
[tex]\displaystyle y=\frac{22- 4}{2}=9[/tex]
The value of x could have two solutions also:
x=22-13=9
x=22-9=13
Since y is the age of the elder brother, the answer is:
The elder brother is 13 years old
