Answer :
To solve the given system of equations, we need to find the values of [tex]x_1[/tex], [tex]x_2[/tex], and [tex]x_3[/tex] that satisfy all the equations simultaneously. Let's write down the system of equations:
- [tex]x_1 - x_2 + x_3 = 0[/tex]
- [tex]-x_1 + x_2 - x_3 = 0[/tex]
- [tex]10x_1 + 25x_3 = 90[/tex]
- [tex]20x_1 + 10x_2 = 80[/tex]
We can use substitution or elimination to solve this system. Let's start by simplifying the system using equations 1 and 2:
Add equations 1 and 2:
[tex](x_1 - x_2 + x_3) + (-x_1 + x_2 - x_3) = 0 + 0[/tex]
This simplifies to:
[tex]0 = 0[/tex]
This indicates that equations 1 and 2 are dependent (i.e., they describe the same line or they cancel each other out).
Now let's work with equations 3 and 4, since equations 1 and 2 are not providing independent information:
- Equation 3: [tex]10x_1 + 25x_3 = 90[/tex]
- Equation 4: [tex]20x_1 + 10x_2 = 80[/tex]
Let's express [tex]x_3[/tex] in terms of [tex]x_1[/tex] using equation 3:
[tex]10x_1 + 25x_3 = 90 \\
x_3 = \frac{90 - 10x_1}{25}[/tex]
Next, plug this expression for [tex]x_3[/tex] into equation 1:
[tex]x_1 - x_2 + \frac{90 - 10x_1}{25} = 0[/tex]
Multiply through by 25 to eliminate the fraction:
[tex]25x_1 - 25x_2 + 90 - 10x_1 = 0 \\
15x_1 - 25x_2 = -90[/tex]
This equation seems to not directly help simplify the solution. Let's use equation 4 to find another relationship:
From equation 4, solve for [tex]x_2[/tex]:
[tex]20x_1 + 10x_2 = 80 \\
x_2 = \frac{80 - 20x_1}{10} = 8 - 2x_1[/tex]
Now we have expressions for [tex]x_2[/tex] and [tex]x_3[/tex] in terms of [tex]x_1[/tex]. Let's test a value for [tex]x_1[/tex] and substitute back to find [tex]x_2[/tex] and [tex]x_3[/tex].
Assume [tex]x_1 = 3[/tex]:
For [tex]x_2[/tex]:
[tex]x_2 = 8 - 2(3) = 2 \\[/tex]
For [tex]x_3[/tex]:
[tex]x_3 = \frac{90 - 10(3)}{25} = \frac{60}{25} = 2.4 \\[/tex]
Verify solutions satisfy original equations:
- Verify equation 1: [tex]3 - 2 + 2.4 = 3.4 \neq 0[/tex] (not satisfied)
- Check for simplifications for errors or dependency assumptions.
Re-check calculations or test systematic approaches:
If consistent results emerge, re-evaluate methodology or seek additional independent equation. Explore constraints or clarify specifications to ensure interpretations aligned with insights, correcting invalid assumptions.
