Find x, y, z, and w if
c
x + y 2x − 3y
z − w z + 2wd = c
 4 −7
−6 6
d

We are given a matrix equation in the form:

\[
\begin{pmatrix} 
x + y & 2x - 3y \\
z - w & z + 2w 
\end{pmatrix}
= 
\begin{pmatrix} 
4 & -7 \\
-6 & 6 
\end{pmatrix}
\]

To solve for \( x \), \( y \), \( z \), and \( w \), we can equate corresponding elements of the matrices. This gives us the following system of equations:

1. \( x + y = 4 \)
2. \( 2x - 3y = -7 \)
3. \( z - w = -6 \)
4. \( z + 2w = 6 \)

Now, let's solve this system of equations.

### Step 1: Solve for \( x \) and \( y \)
From the first two equations:

1. \( x + y = 4 \)
2. \( 2x - 3y = -7 \)

We'll solve this system by substitution or elimination.

### Step 2: Solve for \( z \) and \( w \)
From the last two equations:

3. \( z - w = -6 \)
4. \( z + 2w = 6 \)

We'll solve this system as well.

I'll start by solving for \( x \), \( y \), \( z \), and \( w \).The solutions for the variables are:

- \( x = 1 \)
- \( y = 3 \)
- \( z = -2 \)
- \( w = 4 \)

Would you like any further details or clarifications?

Here are 5 related questions for practice:
1. Solve for \( a \) and \( b \) if \( a + 2b = 10 \) and \( 3a - b = 5 \).
2. If \( 4x + y = 12 \) and \( 2x - 3y = -7 \), solve for \( x \) and \( y \).
3. Find \( z \) and \( w \) when \( z + w = 8 \) and \( z - 2w = -1 \).
4. Solve the system: \( 5p + q = 7 \) and \( 2p - 4q = -10 \).
5. What is the solution for \( x \) and \( y \) if \( 3x + y = 9 \) and \( x - 2y = 3 \)?

**Tip:** For solving systems of linear equations, substitution and elimination methods are effective for reducing the system into a simpler form.

Find x, y, z, and w if cx + y 2x − 3y z − w z + 2wd = c 4 −7 −6 6 d

Solve a matrix equation involving variables x, y, z, and w using systems of equations. This problem demonstrates key concepts in linear algebra, including matrix equality and substitution methods. Suitable for students in Grades 10-12.

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