Certainly! Let's solve for [tex]\( x \)[/tex] step by step using the given information:
Given:
- [tex]\( AB = 10 \)[/tex]
- [tex]\( AC = 2x + 5 \)[/tex]
- [tex]\( BC = 7x - 10 \)[/tex]
Since [tex]\( A \)[/tex] is between [tex]\( B \)[/tex] and [tex]\( C \)[/tex], the distances add up as follows:
[tex]\[ AB + BC = AC \][/tex]
Substitute the given lengths into this equation:
[tex]\[ 10 + (7x - 10) = 2x + 5 \][/tex]
Let's simplify the left-hand side:
[tex]\[ 10 + 7x - 10 = 2x + 5 \][/tex]
[tex]\[ 7x = 2x + 5 \][/tex]
Next, isolate the variable [tex]\( x \)[/tex]:
[tex]\[ 7x - 2x = 5 \][/tex]
[tex]\[ 5x = 5 \][/tex]
Divide both sides by 5:
[tex]\[ x = 1 \][/tex]
Now that we have [tex]\( x = 1 \)[/tex], let's substitute this back into the expressions for [tex]\( AC \)[/tex] and [tex]\( BC \)[/tex] to find their actual lengths:
For [tex]\( AC \)[/tex]:
[tex]\[ AC = 2x + 5 \][/tex]
[tex]\[ AC = 2(1) + 5 \][/tex]
[tex]\[ AC = 2 + 5 \][/tex]
[tex]\[ AC = 7 \][/tex]
For [tex]\( BC \)[/tex]:
[tex]\[ BC = 7x - 10 \][/tex]
[tex]\[ BC = 7(1) - 10 \][/tex]
[tex]\[ BC = 7 - 10 \][/tex]
[tex]\[ BC = -3 \][/tex]
So, the values are:
- [tex]\( x = 1 \)[/tex]
- [tex]\( AC = 7 \)[/tex]
- [tex]\( BC = -3 \)[/tex]
The lengths and the value of [tex]\( x \)[/tex] are as follows:
[tex]\[ x = 1 \][/tex]
[tex]\[ AC = 7 \][/tex]
[tex]\[ BC = -3 \][/tex]
