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Magazine Chapter 4: Problem 30
Solve each equation. $$ \frac{2 t}{2 t^{2}+9 t+10}+\frac{1-3 t}{3 t^{2}+4 t-4}=\frac{4}{6 t^{2}+11 t-10} $$
Short Answer
Expert verified
The solution is \(t = -\frac{1}{7}\).
Step by step solution
01
Identify a common denominator
The equations need to have a common denominator to be combined. The given equation is \( \frac{2t}{2t^2 + 9t + 10} + \frac{1-3t}{3t^2 + 4t - 4} = \frac{4}{6t^2 + 11t - 10} \). Factor each denominator: \( 2t^2 + 9t + 10 = (2t + 5)(t + 2) \), \( 3t^2 + 4t - 4 = (3t - 2)(t + 2) \), and \( 6t^2 + 11t - 10 = (3t - 2)(2t + 5) \). The common denominator is \((2t + 5)(t + 2)(3t - 2)\).
02
Rewrite each term with common denominator
Rewrite each fraction with \((2t + 5)(t + 2)(3t - 2)\) as the common denominator:1. \( \frac{2t}{2t^2 + 9t + 10} \rightarrow \frac{2t(3t - 2)}{(2t + 5)(t + 2)(3t - 2)} \).2. \( \frac{1-3t}{3t^2 + 4t - 4} \rightarrow \frac{(1 - 3t)(2t + 5)}{(3t - 2)(t + 2)(2t + 5)} \).3. \( \frac{4}{6t^2 + 11t - 10} \rightarrow \frac{4(t + 2)}{(3t - 2)(2t + 5)(t + 2)} \).
03
Combine the fractions
Now that all fractions have the same denominator, they can be combined:\(\frac{2t(3t - 2) + (1 - 3t)(2t + 5)}{(2t + 5)(t + 2)(3t - 2)} = \frac{4(t + 2)}{(2t + 5)(t + 2)(3t - 2)}.\)
04
Solve the numerator equation
Distribute and simplify the numerators:1. \( 2t(3t - 2) = 6t^2 - 4t \).2. \( (1 - 3t)(2t + 5) = 2t - 6t^2 + 5 - 15t = -6t^2 - 13t + 5 \).Combine these: \(\frac{6t^2 - 4t + (-6t^2 - 13t + 5)}{(2t + 5)(t + 2)(3t - 2)} = \frac{4(t + 2)}{(2t + 5)(t + 2)(3t - 2)}.\)Now, add and simplify: \(-17t + 5 = 4(t + 2)\).
05
Solve the equation
Rewrite and solve the linear equation: \(-17t + 5 = 4t + 8\).Combine like terms: \(-17t - 4t = 8 - 5\) which becomes \(-21t = 3\). Divide both sides by -21 to find \(t\): \(t = -\frac{1}{7}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring polynomials is like breaking down a complex shape into simpler parts. When you factor a polynomial, you express it as a product of smaller polynomials called factors. In our exercise, we need to factor the denominators of the rational equation.
- Firstly, the polynomial \(2t^2 + 9t + 10\) is factored into \((2t + 5)(t + 2)\). You find these factors by looking for two numbers that multiply to 20 (the product of the coefficient of \(t^2\) and the constant term) and add up to 9, which are 4 and 5. Then, grouping the terms helps to find common factors.
- Similarly, the polynomial \(3t^2 + 4t - 4\) becomes \((3t - 2)(t + 2)\), and \(6t^2 + 11t - 10\) turns into \((3t - 2)(2t + 5)\).
Common Denominator
When solving rational equations, it's important to combine fractions under a common denominator. This process entails equalizing denominators so that we can add or subtract the fractions.Let's consider the rational equation from our example:
- The factored denominators are \((2t + 5)(t + 2)\), \((3t - 2)(t + 2)\), and \((3t - 2)(2t + 5)\).
- We derive a common denominator for all fractions, which is \((2t + 5)(t + 2)(3t - 2)\).
Combining Fractions
Once you have a common denominator, it's time to combine the fractions. This means rewriting each fraction with this denominator and then adding or subtracting the numerators.For the given equation:
- You first express each term with the common denominator: \(\frac{2t(3t - 2)}{(2t + 5)(t + 2)(3t - 2)}\), \(\frac{(1 - 3t)(2t + 5)}{(2t + 5)(t + 2)(3t - 2)}\), and \(\frac{4(t + 2)}{(2t + 5)(t + 2)(3t - 2)}\).
- Then, you combine them to form one fraction: \(\frac{2t(3t - 2) + (1 - 3t)(2t + 5)}{(2t + 5)(t + 2)(3t - 2)} = \frac{4(t + 2)}{(2t + 5)(t + 2)(3t - 2)}\).
Solving Linear Equations
After combining the fractions, the next step is to solve the linear equation. This involves equating the numerators and solving for the variable.
- Start by distributing and simplifying the numerators: \(2t(3t - 2)\) becomes \(6t^2 - 4t\), and \((1 - 3t)(2t + 5)\) results in \(-6t^2 - 13t + 5\).
- Combining these terms, we arrive at \(-17t + 5 = 4(t + 2)\). Simplify further to get the equation \(-17t + 5 = 4t + 8\).
- Rearrange to solve: Combine like terms for \(-21t = 3\) and finally divide by \(-21\) to obtain \(t = -\frac{1}{7}\).
