6. One angle measures 54 more than 3 times another angle. The angles are supplementary. What are their measures?

11. A piece of string is 112 inches long. Isabel wants to cut it into 2 pieces so that one piece is three times as along as the other. How long is each piece?

6. x + 3x + 54º = 180º

11. x + 3x = 112

Solve for x.

45cm

28

To solve these types of problems, we can set up equations based on the given information.

For question 6, let's denote one angle as "x" and the other angle as "y". We know that one angle measures 54 more than 3 times the other angle, so we can write the equation:
x = 3y + 54

We also know that the angles are supplementary, which means their sum is 180 degrees. So, we can write another equation:
x + y = 180

Now, we have a system of equations. We can solve it by substituting the value of x from the first equation into the second equation:
(3y + 54) + y = 180
4y + 54 = 180
4y = 180 - 54
4y = 126
y = 126 / 4
y = 31.5

Now that we have the value of y, we can substitute it back into the first equation to find x:
x = 3y + 54
x = 3(31.5) + 54
x = 94.5 + 54
x = 148.5

So, the two angles measure 148.5 degrees and 31.5 degrees.

Now let's move on to question 11.

Let's denote the length of one piece of string as "x" and the length of the other piece as "y". We know that one piece is three times as long as the other, so we can write the equation:
x = 3y

We also know that the total length of the string is 112 inches, so we can write another equation:
x + y = 112

Now, we have a system of equations. We can solve it by substituting the value of x from the first equation into the second equation:
3y + y = 112
4y = 112
y = 112 / 4
y = 28

Now that we have the value of y, we can substitute it back into the first equation to find x:
x = 3y
x = 3(28)
x = 84

So, one piece of string is 84 inches long and the other piece is 28 inches long.