Problem 24 - Algebra A Combined Function

Question

Solve each equation and check each proposed solution. See Examples 4 through 6. $$ \frac{4 y}{y-3}-3=\frac{3 y-1}{y+3} $$

Step-by-Step Solution

Answer

The solutions are \( y = 12 \) and \( y = -1 \).

Step 1: Identify the Equation Type

The equation provided is a rational equation, as it involves fractions with polynomials in the numerator and denominator: \( \frac{4y}{y-3} - 3 = \frac{3y-1}{y+3} \).

Step 2: Find a Common Denominator

Both denominators \( y-3 \) and \( y+3 \) are different. The common denominator for these two expressions would be \((y-3)(y+3)\).

Step 3: Multiply through by the Common Denominator

Multiply both sides of the equation by \( (y-3)(y+3) \) to eliminate the fractions:1. Left side: \( (y-3)(y+3) \left( \frac{4y}{y-3} - 3 \right) = 4y(y+3) - 3(y+3)(y-3) \).2. Right side: \( (y-3)(y+3) \times \frac{3y-1}{y+3} = (3y-1)(y-3) \).

Step 4: Simplify Both Sides

Simplifying gives:1. Left side: \( 4y^2 + 12y - (3(y^2 - 9)) = 4y^2 + 12y - 3y^2 + 27 \).2. Right side: \( 3y^2 - 9y - y + 3 = 3y^2 - 10y + 3 \).

Step 5: Combine Like Terms

Combine terms on both sides:- Left side: \( (4y^2 - 3y^2) + 12y + 27 = y^2 + 12y + 27 \).- Right side remains: \( 3y^2 - 10y + 3 \).

Step 6: Set Both Sides to a Standard Form

Put the equation in the form of a quadratic equation: \( y^2 + 12y + 27 = 3y^2 - 10y + 3 \) becomes \( 0 = 2y^2 - 22y - 24 \) after moving all terms to one side.

Step 7: Solve the Quadratic Equation

To solve \( 2y^2 - 22y - 24 = 0 \), use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a = 2, b = -22, c = -24 \).

Step 8: Calculate the Discriminant

The discriminant \( b^2 - 4ac \) is calculated as:\((-22)^2 - 4(2)(-24) = 484 + 192 = 676 \).

Step 9: Solve Using the Quadratic Formula

Substitute into the quadratic formula:\{displaystyle y = \frac{22 \pm \sqrt{676}}{4} \}.Ultimately, \( y = \frac{22 \pm 26}{4} \).This gives solutions \( y = 12 \) and \( y = -1 \).

Step 10: Check the Solutions in the Original Equation

Substitute the solutions back into the original equation to verify their validity.- For \( y = 12 \), substituting gives a valid equation balance.- For \( y = -1 \), substituting also shows validity.Hence, both solutions are correct.

Key Concepts

Common Denominator

When dealing with rational equations like \( \frac{4y}{y-3} - 3 = \frac{3y-1}{y+3} \), finding a common denominator is essential. This step ensures we can eliminate fractions and simplify the equation. In this case, the denominators \( y-3 \) and \( y+3 \) differ, so the common denominator is their product, \( (y-3)(y+3) \).
Multiplying the entire equation by this common denominator removes the fractions, making it possible to solve using algebraic methods. This process is crucial because only with a common base can we systematically address all the fraction parts simultaneously. This technique simplifies the equation, paving the way for further operations, such as combining like terms. Remember: finding the right common denominator is like laying a strong foundation for a building—it supports everything that follows.

Quadratic Formula

Once you've achieved the goal of eliminating fractions using a common denominator, your rational equation might become a quadratic one. Quadratic equations have the standard form \( ax^2 + bx + c = 0 \). Our example transformed into \( 2y^2 - 22y - 24 = 0 \) after simplification.
To solve quadratic equations, the quadratic formula is a fundamental tool. It is given by:

• \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula allows us to find the values of \( y \) that satisfy the equation by substituting the coefficients \( a \), \( b \), and \( c \) into the formula. In our example, we found the solutions \( y = 12 \) and \( y = -1 \), showing the formula's power in providing exact solutions to quadratic equations.

Discriminant

The discriminant is a key part of the quadratic formula process. Found within the formula as \( b^2 - 4ac \), it provides information about the nature of the solutions of a quadratic equation. A positive discriminant indicates two real and distinct solutions.
In our exercise, the discriminant was calculated as follows:

• \( (-22)^2 - 4(2)(-24) \)
• which equals \( 676 \)

Since 676 is positive, it confirms that our quadratic equation has two unique solutions. Understanding the discriminant helps to quickly assess what kind of solutions to expect, which is especially vital in checking the validity of your solutions later.

Check Solutions

After solving a rational equation and finding potential solutions, it is critically important to check these solutions in the original equation to ensure they do not introduce any invalid results, such as division by zero or other inconsistencies.
For \( y = 12 \) and \( y = -1 \) found from our quadratic equation, plugging them back into the original equation verifies their correctness. This step confirms that neither solution causes any denominator to become zero and that both actually balance the equation, ensuring each is a valid and acceptable solution. Checking solutions guards against errors arising from algebraic manipulation and reinforces a deeper understanding of rational equations. Always conclude with this process to ensure mathematical accuracy and completeness.