The three angles of a triangle have measures 12x degree, 3x degree, and 7y degree, where 7y>60. If x and y are integers, what is the value of x?
12x + 3x + 7y = 180
15x + 7y = 180
15x + 60 < 180
15x < 120
x < 8
check:
pick a value of x,
1. x < 8 , e.g. let x = 4
then 48+12 + 7y = 180
7y = 120 , so 7y > 60 , OK!
2. x > 8 , e.g. x = 10
then 150 + 30 + 7y = 180
7y = 0 , and 7y > 60 is false
Therefore x < 8 is correct.
To find the value of x, we need to use the fact that the sum of the angles of a triangle is 180 degrees.
Let's write an equation using the given information:
12x + 3x + 7y = 180
Simplifying the equation:
15x + 7y = 180
Now, since 7y is greater than 60, we can see that y must be greater than 8 (7y > 60 implies y > 60/7, so y > 8.57). Since y is an integer, y must be 9 or greater.
Let's substitute y = 9 into the equation:
15x + 7(9) = 180
15x + 63 = 180
Subtracting 63 from both sides of the equation:
15x = 117
Dividing both sides of the equation by 15:
x = 117/15
Simplifying the fraction:
x = 7.8
Since x is an integer, we can conclude that the value of x is 7.
So, the value of x is 7.