Math Problem Statement

Find values for p and q so that x = -2 is the only solution to the equation px + 2 = 4x + q.

Solution

We are tasked with finding the values of pp and qq such that x=2x = -2 is the only solution to the equation px+2=4x+qpx + 2 = 4x + q.

Step-by-step Solution:

  1. Substitute x=2x = -2 into the equation: Since x=2x = -2 is the solution, we plug x=2x = -2 into the equation px+2=4x+qpx + 2 = 4x + q.

    Substituting x=2x = -2: p(2)+2=4(2)+qp(-2) + 2 = 4(-2) + q Simplifying both sides: 2p+2=8+q-2p + 2 = -8 + q

  2. Solve for qq in terms of pp: To find the relationship between pp and qq, solve the equation: 2p+2=8+q-2p + 2 = -8 + q Rearranging: q=2p+10q = -2p + 10

  3. Ensure that x=2x = -2 is the only solution: For x=2x = -2 to be the only solution, the coefficients of xx on both sides of the original equation must be equal. This means pp must equal 44, so the equation doesn't have multiple solutions.

    Therefore, set p=4p = 4.

  4. Substitute p=4p = 4 back into the equation for qq: q=2(4)+10=8+10=2q = -2(4) + 10 = -8 + 10 = 2

Final Answer:

p=4,q=2p = 4, \quad q = 2

Would you like more details on any step, or have any other questions?


Here are 5 related questions to explore:

  1. What happens if p4p \neq 4? Will the equation have more than one solution?
  2. How would the solution change if the condition was that x=3x = 3 is the only solution?
  3. How can you check if a given equation has exactly one solution algebraically?
  4. What is the general method for solving a system of linear equations like this one?
  5. Can we solve this problem geometrically by interpreting the lines represented by the equations?

Tip: Always check if the equation is linear in order to guarantee a single solution when applying such conditions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

Linear equation px + c = qx + d

Theorems

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Suitable Grade Level

Grades 7-9