Teach me step by step how to solve XY=11d, YZ=9d−2, XZ=5d+28

To solve the system of equations:

1. \( XY = 11d \)
2. \( YZ = 9d - 2 \)
3. \( XZ = 5d + 28 \)

where \( X \), \( Y \), and \( Z \) are variables, and \( d \) is a known value, follow these steps:

### Step 1: Express \( X \), \( Y \), and \( Z \) in terms of each other.

You have three equations, and we will first try to isolate one of the variables.

#### From Equation 1: \( XY = 11d \)
You can express \( X \) as:
\[
X = \frac{11d}{Y}
\]
or express \( Y \) as:
\[
Y = \frac{11d}{X}
\]

#### From Equation 2: \( YZ = 9d - 2 \)
You can express \( Z \) as:
\[
Z = \frac{9d - 2}{Y}
\]
or express \( Y \) as:
\[
Y = \frac{9d - 2}{Z}
\]

#### From Equation 3: \( XZ = 5d + 28 \)
You can express \( X \) as:
\[
X = \frac{5d + 28}{Z}
\]
or express \( Z \) as:
\[
Z = \frac{5d + 28}{X}
\]

### Step 2: Substitute and solve

Let’s choose one equation to substitute into the others.

#### Substituting \( Z = \frac{9d - 2}{Y} \) into \( XZ = 5d + 28 \):
\[
X \times \frac{9d - 2}{Y} = 5d + 28
\]
Multiply both sides by \( Y \) to eliminate the fraction:
\[
X(9d - 2) = (5d + 28)Y
\]
Now we can express \( X \) from Equation 1 as \( X = \frac{11d}{Y} \), and substitute it here:
\[
\frac{11d(9d - 2)}{Y} = (5d + 28)Y
\]
Now multiply both sides by \( Y \) to get rid of the denominator:
\[
11d(9d - 2) = (5d + 28)Y^2
\]
Simplify both sides:
\[
11d(9d - 2) = 99d^2 - 22d
\]
So the equation becomes:
\[
99d^2 - 22d = (5d + 28)Y^2
\]

### Step 3: Solve the system

Now, to proceed further, you can either isolate \( Y^2 \) and solve for \( Y \), or simplify further to find relationships between \( X \), \( Y \), and \( Z \). Depending on the values of \( d \), we can substitute and solve.

Would you like to continue with specific values for \( d \) or simplify the equations further?

---

**Questions:**
1. Do you want to proceed with solving for \( Y \), or would you like to solve for a different variable?
2. Would you like to substitute a specific value of \( d \) into the equations?
3. Do you want to see the equations simplified step by step for \( X \), \( Y \), and \( Z \)?
4. Are you comfortable with quadratic equations to solve for \( Y \)?
5. Would you prefer solving the system graphically?

**Tip:** When dealing with multiple equations, always try to reduce the number of variables by substitution. This makes the system easier to manage.

Explore how to solve a system of equations involving variables and constants step by step. This detailed guide covers algebraic substitution methods and their applications to equations. Suitable for high school students (Grades 10-12).

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