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:
Since point is on the line segment , the total length of the segment can be written as: Substituting the given expressions for , , and , we have: Now, simplify the right-hand side: Next, solve for by moving the terms involving to one side and the constants to the other:
Now that we have the value of , substitute it back into the expression for :
Thus, the numerical length of is .
Would you like further details or have any questions?
Here are 5 related questions to extend your understanding:
- What would the length of be if ?
- How does changing affect the lengths of and ?
- How would you verify that with ?
- If the length of were increased, how would it affect the value of ?
- What if the equation for was different, say ? 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
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