Math Problem Statement

Add. Simplify by removing a factor of 1 when possible: (7z - 11)/(z^2 - 7z + 12) + (7 - 6z)/(z^2 - 7z + 12).

Solution

To solve and simplify the given problem:

7z11z27z+12+76zz27z+12\frac{7z - 11}{z^2 - 7z + 12} + \frac{7 - 6z}{z^2 - 7z + 12}


Step 1: Common Denominator

Both fractions already have the same denominator: z27z+12z^2 - 7z + 12. So we can combine the numerators directly:

(7z11)+(76z)z27z+12.\frac{(7z - 11) + (7 - 6z)}{z^2 - 7z + 12}.


Step 2: Combine the Numerators

Simplify the numerator (7z11)+(76z)(7z - 11) + (7 - 6z):

  1. Combine like terms: 7z6z+(11)+7=z4.7z - 6z + (-11) + 7 = z - 4.

Thus, the numerator simplifies to z4z - 4.


Step 3: Simplify the Expression

The simplified fraction is:

z4z27z+12.\frac{z - 4}{z^2 - 7z + 12}.

Now, factor the denominator z27z+12z^2 - 7z + 12. This quadratic can be factored as:

z27z+12=(z3)(z4).z^2 - 7z + 12 = (z - 3)(z - 4).


Step 4: Cancel Common Factors

The numerator z4z - 4 and one factor in the denominator z4z - 4 cancel out (as long as z4z \neq 4, to avoid division by zero):

z4(z3)(z4)=1z3.\frac{z - 4}{(z - 3)(z - 4)} = \frac{1}{z - 3}.


Final Simplified Answer:

1z3,where z4 or 3.\frac{1}{z - 3}, \quad \text{where } z \neq 4 \text{ or } 3.


Would you like a detailed explanation of any specific step? Let me know!

Related Questions:

  1. How do you factor a quadratic expression like z27z+12z^2 - 7z + 12?
  2. What are the restrictions on the variable zz when simplifying rational expressions?
  3. How do you combine fractions with a common denominator?
  4. What happens when terms in the numerator and denominator cancel?
  5. How can you check your final simplified expression for correctness?

Tip:

Always check for restrictions (values that make the denominator zero) when simplifying rational expressions!

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Math Problem Analysis

Mathematical Concepts

Rational Expressions
Factoring Polynomials
Simplifying Fractions

Formulas

Combining fractions: a/b + c/b = (a + c)/b
Factoring quadratic equations: z^2 - 7z + 12 = (z - 3)(z - 4)

Theorems

Cancellation Property of Rational Expressions

Suitable Grade Level

Grades 9-11