Find the value of a, b, c, and d. Enter all answers in exact, reduced form. No decimals are allowed. Any negative signs must be in the numerator.

7	8

d	−4
T − 5

−4	−4

−c	5
= −8

a	5

−3	b
a =
Incorrect: Your answer is incorrect.
b =
Incorrect: Your answer is incorrect.
c =
Incorrect: Your answer is incorrect.
d =
Correct: Your answer is correct.

Let's analyze the given system of equations. From the layout, it seems like we're dealing with a matrix equation or a set of related equalities. However, the information provided is a bit ambiguous in format. I'll assume that each fraction represents an equation comparing two ratios. 

Here’s what the problem looks like:

\[
\frac{7}{d} = \frac{8}{-4}
\]

\[
\frac{-4}{-c} = \frac{-4}{5}
\]

\[
\frac{-8}{a} = \frac{5}{-3}
\]

\[
\frac{a}{b} = \frac{T-5}{T-5}
\]

The above equations can be solved for \(a\), \(b\), \(c\), and \(d\):

### Equation 1: \(\frac{7}{d} = \frac{8}{-4}\)
\[
\frac{7}{d} = -2 \Rightarrow d = \frac{7}{-2} = -\frac{7}{2}
\]
So, \(d = -\frac{7}{2}\).

### Equation 2: \(\frac{-4}{-c} = \frac{-4}{5}\)
\[
\frac{-4}{-c} = \frac{-4}{5} \Rightarrow \frac{4}{c} = \frac{-4}{5} \Rightarrow 4c = -20 \Rightarrow c = -5
\]
So, \(c = -5\).

### Equation 3: \(\frac{-8}{a} = \frac{5}{-3}\)
\[
\frac{-8}{a} = -\frac{5}{3} \Rightarrow -8 \times -3 = 5a \Rightarrow 24 = 5a \Rightarrow a = \frac{24}{5}
\]
So, \(a = \frac{24}{5}\).

### Equation 4: \(\frac{a}{b} = \frac{T-5}{T-5}\)
Since \(\frac{T-5}{T-5} = 1\), we have:
\[
\frac{a}{b} = 1 \Rightarrow a = b
\]
So, \(b = \frac{24}{5}\).

Thus, the solutions are:
- \(a = \frac{24}{5}\)
- \(b = \frac{24}{5}\)
- \(c = -5\)
- \(d = -\frac{7}{2}\)

Would you like further details on any of the steps? Here are five related questions for further exploration:

1. How do you solve a system of linear equations using matrix methods?
2. What are the conditions for two ratios to be equal?
3. How can you check the consistency of a solution in a system of equations?
4. What methods can be used to solve non-linear systems of equations?
5. How does the determinant of a matrix affect the solution to a system of equations?

**Tip:** When solving equations involving fractions, cross-multiply to eliminate the fractions and simplify the process.

Learn how to solve equations involving variables and fractions with detailed step-by-step solutions. This problem covers key concepts like cross-multiplication and reducing fractions to their simplest form. Suitable for high school students.

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