Certainly! We are given two equations involving three variables [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex]:
1. [tex]\( 5x - 2y = 11 \)[/tex]
2. [tex]\( 4y - 5z = -17 \)[/tex]
Let's solve this system step by step.
### Step 1: Isolate one variable in terms of the others from one of the equations
Let's start with the first equation [tex]\( 5x - 2y = 11 \)[/tex]. We can isolate [tex]\( x \)[/tex]:
[tex]\[ 5x = 2y + 11 \][/tex]
[tex]\[ x = \frac{2y + 11}{5} \][/tex]
### Step 2: Substitute into the second equation
Next, let's solve the second equation [tex]\( 4y - 5z = -17 \)[/tex] for [tex]\( y \)[/tex]:
[tex]\[ 4y = 5z - 17 \][/tex]
[tex]\[ y = \frac{5z - 17}{4} \][/tex]
### Step 3: Express [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in terms of [tex]\( z \)[/tex]
Now that we have [tex]\( y \)[/tex] in terms of [tex]\( z \)[/tex], we can substitute [tex]\( y = \frac{5z - 17}{4} \)[/tex] into the expression we found for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2y + 11}{5} \][/tex]
[tex]\[ x = \frac{2 \left( \frac{5z - 17}{4} \right) + 11}{5} \][/tex]
Simplify the fraction inside the parentheses:
[tex]\[ x = \frac{\frac{2(5z - 17)}{4} + 11}{5} \][/tex]
[tex]\[ x = \frac{\frac{10z - 34}{4} + 11}{5} \][/tex]
Find a common denominator to combine the terms inside the fraction:
[tex]\[ x = \frac{\frac{10z - 34 + 44}{4}}{5} \][/tex]
[tex]\[ x = \frac{\frac{10z + 10}{4}}{5} \][/tex]
[tex]\[ x = \frac{10(z + 1)}{4 \cdot 5} \][/tex]
[tex]\[ x = \frac{10(z + 1)}{20} \][/tex]
[tex]\[ x = \frac{z + 1}{2} \][/tex]
Thus, [tex]\( x = \frac{z}{2} + \frac{1}{2} \)[/tex].
### Step 4: Collect the final results
So, we have expressed both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in terms of [tex]\( z \)[/tex]:
[tex]\[ x = \frac{z}{2} + \frac{1}{2} \][/tex]
[tex]\[ y = \frac{5z - 17}{4} \][/tex]
Thus, the solution set for the given system of equations is:
[tex]\[ \boxed{x = \frac{z}{2} + \frac{1}{2}, \quad y = \frac{5z - 17}{4}} \][/tex]
These expressions describe [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in terms of [tex]\( z \)[/tex], providing a parametric solution to the system of equations.
