Problem 32 Solve. If no solution exists, st... [FREE SOLUTION]

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Chapter 7: Problem 32

Solve. If no solution exists, state this. $$\frac{t}{t-6}=\frac{36}{t^{2}-6 t}$$

Short Answer

Expert verified

The solution is $t = -6$.

Step by step solution

- Identify the Equation

The given equation is $\frac{t}{t-6}=\frac{36}{t^{2}-6t}$.

- Factor the Denominator

Factor the denominator on the right-hand side: $\frac{t}{t-6}=\frac{36}{t(t-6)}$.

- Simplify the Equation

Notice that both sides have a common term $t-6$. Simplify the equation: $t = \frac{36}{t}$.

- Solve the Simplified Equation

Multiply both sides by $t$ to get: $t^2 = 36$.

- Find the Roots

Solve the quadratic equation: $t^2 = 36$. The solutions are $t = 6$ and $t = -6$.

- Check for Exclusions

Check if any values are excluded due to the original denominator $t-6=0$ which makes $t eq 6$. Since $t = 6$ is excluded, the only solution is $t = -6$.

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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 a key skill when working with rational equations. Polynomials can often be factored into simpler expressions that make solving equations easier. In our exercise, factor the polynomial in the denominator:
$t^2 - 6t$
is rewritten as
$t(t-6)$
. Factoring helps to reveal common terms which can simplify the equation. When you spot a common factor in both the numerator and the denominator, you can often reduce the equation to a much simpler form. This technique is crucial in unraveling complex rational expressions.

Simplifying Equations

Simplifying equations involves reducing them to their most basic form. The goal is to make equations easier to solve by spotting and canceling common terms. In the given problem, you could subtract or divide common terms found in both the numerator and the denominator on both sides of the equation. Once we factor the denominator from
\frac{36}{{t(t-6)}}
, to
\fraction{36}{t(t-6)}
, we see that $t-6$ can be canceled on both sides since it is a common factor. Simplifying brings us to $t = \frac{36}{t}$. This new form is much easier to handle than the initial equation.

Quadratic Equations

Understanding quadratic equations is vital for solving rational equations. A quadratic equation typically takes the form
$ax^2 + bx + c = 0$
. When we simplify our problem, we end up with $t^2 = 36$. This equation is quadratic. To solve it, we take the square root of both sides, resulting in $t = 6$ and $t = -6$. Don鈥檛 forget to check all solutions in the original equation to ensure they do not make any denominators zero. Here, we exclude $t = 6$ because it causes division by zero in the original denominator. Thus, the only viable solution is $t = -6$. Recognizing which values can and cannot be solutions is critical for accurate problem-solving.

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91影视

Solve. If no solution exists, state this. $$\frac{t}{t-6}=\frac{36}{t^{2}-6 t}$$

Short Answer

Step by step solution

- Identify the Equation

- Factor the Denominator

- Simplify the Equation

- Solve the Simplified Equation

- Find the Roots

- Check for Exclusions

Key Concepts

Factoring Polynomials

Simplifying Equations

Quadratic Equations

One App. One Place for Learning.

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