OP = 4x-5, NP = 3x+4 NO = 2x-3 determine the numerical length of OP

To determine the numerical length of \( OP \), we need to use the relationship between the three given line segments \( OP \), \( NP \), and \( NO \).

Given:
- \( OP = 4x - 5 \)
- \( NP = 3x + 4 \)
- \( NO = 2x - 3 \)

Assuming this is a geometric line and \( NO = OP + NP \), we can set up the equation:

\[
NO = OP + NP
\]

Substitute the given expressions:

\[
2x - 3 = (4x - 5) + (3x + 4)
\]

Simplify the right-hand side:

\[
2x - 3 = 4x - 5 + 3x + 4
\]
\[
2x - 3 = 7x - 1
\]

Now, solve for \( x \):

\[
2x - 7x = -1 + 3
\]
\[
-5x = 2
\]
\[
x = -\frac{2}{5}
\]

Now that we have \( x = -\frac{2}{5} \), substitute this value back into the expression for \( OP \) to find its numerical length:

\[
OP = 4x - 5 = 4\left(-\frac{2}{5}\right) - 5
\]
\[
OP = -\frac{8}{5} - 5 = -\frac{8}{5} - \frac{25}{5} = -\frac{33}{5}
\]

Thus, the numerical length of \( OP \) is \( -\frac{33}{5} \) or approximately \( -6.6 \).

Since a length cannot be negative, the absolute value of this is \( \frac{33}{5} \) or 6.6 units.

### Do you want more details on this, or do you have any questions?

Here are some related questions you might find useful:
1. How can you verify the result using another method?
2. What happens if the segments do not form a straight line?
3. What is the relationship between the values of \( NP \) and \( OP \) in terms of \( x \)?
4. How would this change if \( NO \) was greater than \( OP + NP \)?
5. How can we interpret the negative value of \( x \) geometrically?

**Tip:** When solving linear equations involving geometric segments, double-check whether they need to be added or subtracted depending on their alignment on the line.

OP = 4x - 5, NP = 3x + 4, NO = 2x - 3. Determine the numerical length of OP.

Solve the problem of finding the numerical length of OP given the linear equations OP = 4x - 5, NP = 3x + 4, and NO = 2x - 3. This step-by-step solution covers algebraic techniques for solving linear equations and determining segment lengths. Suitable for Grades 8-10.

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