Math Problem Statement

Point O is on line segment start overline, N, P, end overline NP . Given O, P, equals, 4, x, minus, 5, commaOP=4x−5, N, P, equals, 3, x, plus, 4, commaNP=3x+4, and N, O, equals, 2, x, minus, 3, commaNO=2x−3, determine the numerical length of start overline, O, P, end overline, . OP

Solution

We are given the following information:

  • OP=4x5OP = 4x - 5
  • NP=3x+4NP = 3x + 4
  • NO=2x3NO = 2x - 3

Since point OO is on the line segment NPNP, the total length of the segment NPNP can be written as: NP=NO+OPNP = NO + OP Substituting the given expressions for NONO, OPOP, and NPNP, we have: 3x+4=(2x3)+(4x5)3x + 4 = (2x - 3) + (4x - 5) Now, simplify the right-hand side: 3x+4=2x3+4x53x + 4 = 2x - 3 + 4x - 5 3x+4=6x83x + 4 = 6x - 8 Next, solve for xx by moving the terms involving xx to one side and the constants to the other: 3x6x=843x - 6x = -8 - 4 3x=12-3x = -12 x=4x = 4

Now that we have the value of xx, substitute it back into the expression for OPOP: OP=4x5=4(4)5=165=11OP = 4x - 5 = 4(4) - 5 = 16 - 5 = 11

Thus, the numerical length of OPOP is 11\boxed{11}.

Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:

  1. What would the length of NPNP be if x=4x = 4?
  2. How does changing xx affect the lengths of NONO and OPOP?
  3. How would you verify that NO+OP=NPNO + OP = NP with x=4x = 4?
  4. If the length of NPNP were increased, how would it affect the value of xx?
  5. What if the equation for NONO was different, say NO=3x5NO = 3x - 5? How would that change the solution?

Tip: When solving algebraic problems, always double-check your final answer by substituting values back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Geometry (Line Segments)

Formulas

NP = NO + OP
Equation simplification and solving: ax + b = cx + d

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 8-10