Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
