Alright class, let’s address Problem 21 where we need to find the lengths of [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex] given certain equations in terms of [tex]\( x \)[/tex].
First, we are given the following equations:
[tex]\[ AC = 8x - 16 \][/tex]
[tex]\[ AB = 3x - 8 \][/tex]
[tex]\[ BC = 4x \][/tex]
Since point [tex]\( B \)[/tex] is between points [tex]\( A \)[/tex] and [tex]\( C \)[/tex] on the line segment [tex]\(\overline{AC}\)[/tex], we know that:
[tex]\[ AB + BC = AC \][/tex]
Plugging in the given expressions:
[tex]\[ (3x - 8) + 4x = 8x - 16 \][/tex]
Combine like terms on the left side:
[tex]\[ 3x - 8 + 4x = 8x - 16 \][/tex]
[tex]\[ 7x - 8 = 8x - 16 \][/tex]
Now, subtract [tex]\( 7x \)[/tex] from both sides to isolate the term with [tex]\( x \)[/tex] on one side:
[tex]\[ -8 = x - 16 \][/tex]
Add 16 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ 8 = x \][/tex]
So, we have found the value of [tex]\( x \)[/tex]:
[tex]\[ x = 8 \][/tex]
Now let's find the lengths of [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex] by substituting [tex]\( x = 8 \)[/tex] back into the original expressions.
Substitute [tex]\( x = 8 \)[/tex] into [tex]\( AB \)[/tex]:
[tex]\[ AB = 3x - 8 \][/tex]
[tex]\[ AB = 3(8) - 8 \][/tex]
[tex]\[ AB = 24 - 8 \][/tex]
[tex]\[ AB = 16 \][/tex]
Substitute [tex]\( x = 8 \)[/tex] into [tex]\( BC \)[/tex]:
[tex]\[ BC = 4x \][/tex]
[tex]\[ BC = 4(8) \][/tex]
[tex]\[ BC = 32 \][/tex]
Substitute [tex]\( x = 8 \)[/tex] into [tex]\( AC \)[/tex]:
[tex]\[ AC = 8x - 16 \][/tex]
[tex]\[ AC = 8(8) - 16 \][/tex]
[tex]\[ AC = 64 - 16 \][/tex]
[tex]\[ AC = 48 \][/tex]
Therefore, the lengths are:
[tex]\[ AB = 16 \][/tex]
[tex]\[ BC = 32 \][/tex]
[tex]\[ AC = 48 \][/tex]
