Question

Describe and compare the solution sets of x1 - 3x2 + 5x3 = 0 and X1 3x2 + 5x3 = 4. Hint: You must solve two problems each of which has a single equation, which in turn represents a plane. For both, only x is the basic variable. เวไนย ขจรศก

Answer 1

To describe and compare the solution sets of the equations x1 - 3x2 + 5x3 = 0 and x1 + 3x2 + 5x3 = 4, we need to solve each equation separately. The solution set for the first equation is (x1, x2, x3) = (2 - 5x3, 2/3, x3), and the solution set for the second equation is (x1, x2, x3) = (-1/2, 2/3, 1/2).

Step-by-step explanation:

To describe and compare the solution sets of the equations x1 - 3x2 + 5x3 = 0 and x1 + 3x2 + 5x3 = 4, we need to solve each equation separately. Let's start with the first equation:

x1 - 3x2 + 5x3 = 0

Here, x1 is the basic variable and x2 and x3 are parameters. We can solve for x1 by expressing it in terms of x2 and x3:

x1 = 3x2 - 5x3

Now let's solve the second equation:

x1 + 3x2 + 5x3 = 4

Since we know the expression for x1 from the first equation, we can substitute it in the second equation:

3x2 - 5x3 + 3x2 + 5x3 = 4

Combining like terms, we get:

6x2 = 4

Dividing both sides by 6, we find:

x2 = 4/6 = 2/3

Now we can substitute the value of x2 into the expression for x1:

x1 = 3(2/3) - 5x3

x1 = 2 - 5x3

So the solution set for the first equation is:

(x1, x2, x3) = (2 - 5x3, 2/3, x3)

For the second equation, we have:

x1 + 3x2 + 5x3 = 4

Using the expression for x1 from the first equation:

(2 - 5x3) + 3(2/3) + 5x3 = 4

Simplifying the equation, we get:

2 - 3 + 2 + 5x3 = 4

6x3 = 3

Dividing both sides by 6, we find:

x3 = 3/6 = 1/2

Now we can substitute the value of x3 into the expression for x1:

x1 = 2 - 5(1/2)

x1 = 2 - 5/2 = 4/2 - 5/2 = -1/2

So the solution set for the second equation is:

(x1, x2, x3) = (-1/2, 2/3, 1/2) .