Answer :
To solve this system of linear equations, we aim to find the values of [tex]\( x_1 \)[/tex], [tex]\( x_2 \)[/tex], [tex]\( x_3 \)[/tex], and [tex]\( x_4 \)[/tex] that satisfy all four equations simultaneously. Here is a step-by-step explanation of the solution:
1. Equations Recap:
We have the following system of equations:
[tex]\[
\begin{align*}
&1. \quad 3x_1 - 8x_2 + 4x_3 = 19 \\
&2. \quad -6x_1 + 2x_3 + 4x_4 = 5 \\
&3. \quad 5x_1 + 22x_3 + x_4 = 29 \\
&4. \quad x_1 - 2x_2 + 2x_3 = 8 \\
\end{align*}
\][/tex]
2. Approach:
To solve this set of equations, we use the method of substitution or elimination. Here we approach it through conceptual steps that lead to solving such a system.
3. Solving Steps:
- Express [tex]\(x_1\)[/tex] in terms of other variables from Equation 4:
From the fourth equation:
[tex]\[
x_1 = 2x_2 - 2x_3 + 8
\][/tex]
- Substitute into other equations:
Replace [tex]\(x_1\)[/tex] in other equations to reduce variables step-by-step:
For instance, substitute [tex]\(x_1 = 2x_2 - 2x_3 + 8\)[/tex] into Equation 1 and Equation 3 to express other variables similarly.
- Simplify and reduce:
After each substitution, simplify the equations to fewer variables, making them easier to solve in subsequent steps.
- Solve for one variable at a time:
Rearrange the reduced equations to isolate one variable. This can be tedious by hand, but conceptually, you continue this process until you solve for each variable one by one.
4. Numerical Solution:
Following such reductions and substitutions, you end up with numerical values for the variables that solve the system:
- [tex]\( x_1 = -\frac{337}{9} \)[/tex]
- [tex]\( x_2 = -\frac{91}{9} \)[/tex]
- [tex]\( x_3 = \frac{227}{18} \)[/tex]
- [tex]\( x_4 = -\frac{551}{9} \)[/tex]
These are the correct values that satisfy all the given equations.
1. Equations Recap:
We have the following system of equations:
[tex]\[
\begin{align*}
&1. \quad 3x_1 - 8x_2 + 4x_3 = 19 \\
&2. \quad -6x_1 + 2x_3 + 4x_4 = 5 \\
&3. \quad 5x_1 + 22x_3 + x_4 = 29 \\
&4. \quad x_1 - 2x_2 + 2x_3 = 8 \\
\end{align*}
\][/tex]
2. Approach:
To solve this set of equations, we use the method of substitution or elimination. Here we approach it through conceptual steps that lead to solving such a system.
3. Solving Steps:
- Express [tex]\(x_1\)[/tex] in terms of other variables from Equation 4:
From the fourth equation:
[tex]\[
x_1 = 2x_2 - 2x_3 + 8
\][/tex]
- Substitute into other equations:
Replace [tex]\(x_1\)[/tex] in other equations to reduce variables step-by-step:
For instance, substitute [tex]\(x_1 = 2x_2 - 2x_3 + 8\)[/tex] into Equation 1 and Equation 3 to express other variables similarly.
- Simplify and reduce:
After each substitution, simplify the equations to fewer variables, making them easier to solve in subsequent steps.
- Solve for one variable at a time:
Rearrange the reduced equations to isolate one variable. This can be tedious by hand, but conceptually, you continue this process until you solve for each variable one by one.
4. Numerical Solution:
Following such reductions and substitutions, you end up with numerical values for the variables that solve the system:
- [tex]\( x_1 = -\frac{337}{9} \)[/tex]
- [tex]\( x_2 = -\frac{91}{9} \)[/tex]
- [tex]\( x_3 = \frac{227}{18} \)[/tex]
- [tex]\( x_4 = -\frac{551}{9} \)[/tex]
These are the correct values that satisfy all the given equations.
