Math Problem Statement

y = 5 / [(3x + 1)(7x - 2)]

Solution

The given equation from the image is:

y=5(3x+1)(7x2)y = \frac{5}{(3x + 1)(7x - 2)}

This is a rational function where yy is expressed as the quotient of a constant 5 and the product of two linear expressions, (3x+1)(3x + 1) and (7x2)(7x - 2).

Key Aspects:

  • Numerator: 55 is a constant.
  • Denominator: The denominator is the product of two factors, (3x+1)(3x + 1) and (7x2)(7x - 2), both of which are linear terms in xx.

To explore the properties of this function further:

  1. Domain: The domain excludes values of xx where the denominator equals zero. We solve: (3x+1)(7x2)=0(3x + 1)(7x - 2) = 0 This gives: 3x+1=0or7x2=03x + 1 = 0 \quad \text{or} \quad 7x - 2 = 0 Solving for xx: x=13orx=27x = -\frac{1}{3} \quad \text{or} \quad x = \frac{2}{7} So, the domain of yy is all real numbers except x=13x = -\frac{1}{3} and x=27x = \frac{2}{7}.

  2. Vertical Asymptotes:

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Domain and Asymptotes

Formulas

y = 5 / [(3x + 1)(7x - 2)]

Theorems

Vertical Asymptotes
Domain Restrictions

Suitable Grade Level

Grades 9-12