Find x, y, z, and w.

x	9	(2x − 1)

y	7	4y
=

(2x − 5)	z	9

5	(w + 1)	(3y + 5)

We are given the following matrix equation involving variables \(x\), \(y\), \(z\), and \(w\):

\[
\begin{pmatrix}
x & 9 & (2x - 1) \\
y & 7 & 4y \\
(2x - 5) & z & 9 \\
5 & (w + 1) & (3y + 5)
\end{pmatrix}
=
\begin{pmatrix}
x & 9 & (2x - 1) \\
y & 7 & 4y \\
(2x - 5) & z & 9 \\
5 & (w + 1) & (3y + 5)
\end{pmatrix}
\]

### Solution Process:

This is a matrix equality problem. In matrix equations, corresponding entries from both sides of the equation must be equal. So, we can match elements from the matrices and create the system of equations to solve for \(x\), \(y\), \(z\), and \(w\).

### Step 1: Compare elements from the first row
\[
x = x,\quad 9 = 9,\quad (2x - 1) = (2x - 1)
\]
No new information here.

### Step 2: Compare elements from the second row
\[
y = y,\quad 7 = 7,\quad 4y = 4y
\]
No new information here either.

### Step 3: Compare elements from the third row
\[
(2x - 5) = (2x - 5),\quad z = z,\quad 9 = 9
\]
Again, no new information.

### Step 4: Compare elements from the fourth row
\[
5 = 5,\quad (w + 1) = (w + 1),\quad (3y + 5) = (3y + 5)
\]
No new equations to solve.

Thus, this system does not give us any new information or contradictions; it is trivially satisfied for all values of \(x\), \(y\), \(z\), and \(w\).

### Final Answer:
The variables \(x\), \(y\), \(z\), and \(w\) can be any real numbers.

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Here are 5 related questions for practice:
1. How do we solve systems of linear equations using matrices?
2. What does it mean when two matrices are equal?
3. How can we use matrix operations to solve systems of equations?
4. What are the conditions for matrix equality in general?
5. Can matrices of different dimensions ever be equal?

**Tip:** When solving systems using matrices, ensure that corresponding elements in equal matrices are compared properly for consistency.

Find x, y, z, and w.

x 9 (2x − 1)
y 7 4y = (2x − 5) z 9
5 (w + 1) (3y + 5)

Learn how to solve for variables x, y, z, and w by using matrix equality. This problem involves key linear algebra concepts like comparing corresponding matrix elements to solve systems of equations. Suitable for students in Grades 10-12.

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y 7 4y

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