To solve the given system of linear equations:
1. [tex]\(x + y + 3z = 10\)[/tex]
2. [tex]\(4x + 2y + 5x = 7\)[/tex]
3. [tex]\(kx + z = 3\)[/tex]
Let's go through this step-by-step:
### Step 1: Simplify the Second Equation
First, simplify the second equation:
[tex]\[ 4x + 2y + 5x = 7 \][/tex]
[tex]\[ 9x + 2y = 7 \][/tex]
### Step 2: Solve for [tex]\( y \)[/tex] in Terms of [tex]\( x \)[/tex] and [tex]\( z \)[/tex] Using the First Equation
Rearrange the first equation:
[tex]\[ x + y + 3z = 10 \][/tex]
[tex]\[ y = 10 - x - 3z \][/tex]
### Step 3: Substitute [tex]\( y \)[/tex] into the Simplified Second Equation
Substitute [tex]\( y = 10 - x - 3z \)[/tex] into [tex]\( 9x + 2y = 7 \)[/tex]:
[tex]\[ 9x + 2(10 - x - 3z) = 7 \][/tex]
[tex]\[ 9x + 20 - 2x - 6z = 7 \][/tex]
[tex]\[ 7x - 6z = -13 \][/tex]
[tex]\[ 7x = 6z - 13 \][/tex]
[tex]\[ x = \frac{6z - 13}{7} \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] Back into [tex]\( y = 10 - x - 3z \)[/tex]
Substitute [tex]\( x = \frac{6z - 13}{7} \)[/tex] into [tex]\( y = 10 - x - 3z \)[/tex]:
[tex]\[ y = 10 - \frac{6z - 13}{7} - 3z \][/tex]
[tex]\[ y = 10 - \frac{6z - 13}{7} - \frac{21z}{7} \][/tex]
[tex]\[ y = 10 - \frac{6z - 13 + 21z}{7} \][/tex]
[tex]\[ y = 10 - \frac{27z - 13}{7} \][/tex]
[tex]\[ y = \frac{70 - (27z - 13)}{7} \][/tex]
[tex]\[ y = \frac{70 - 27z + 13}{7} \][/tex]
[tex]\[ y = \frac{83 - 27z}{7} \][/tex]
[tex]\[ y = -\frac{27z - 83}{7} \][/tex]
### Step 5: Use the Third Equation to Solve for [tex]\( k \)[/tex]
Substitute [tex]\( x = \frac{6z - 13}{7} \)[/tex] and [tex]\( z = z \)[/tex] into the third equation:
[tex]\[ kx + z = 3 \][/tex]
[tex]\[ k(\frac{6z - 13}{7}) + z = 3 \][/tex]
[tex]\[ \frac{k(6z - 13)}{7} + z = 3 \][/tex]
[tex]\[ \frac{6kz - 13k + 7z}{7} = 3 \][/tex]
[tex]\[ 6kz - 13k + 7z = 21 \][/tex]
[tex]\[ z(6k + 7) = 21 + 13k \][/tex]
[tex]\[ k = \frac{7(3 - z)}{6z - 13} \][/tex]
[tex]\[ k = -\frac{7(z - 3)}{6z - 13} \][/tex]
### Final Solution
The final solutions for the variables are:
[tex]\[ x = \frac{6z - 13}{7} \][/tex]
[tex]\[ y = -\frac{27z - 83}{7} \][/tex]
[tex]\[ z = z \][/tex]
[tex]\[ k = -\frac{7(z - 3)}{6z - 13} \][/tex]
Thus, the solutions can be written as:
[tex]\[
\left( \frac{6z - 13}{7}, -\frac{27z - 83}{7}, z, -\frac{7(z - 3)}{6z - 13} \right)
\][/tex]
