Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.)
2x1	 + 	x2	 − 	3x3	 = 	4
4x1			 + 	2x3	 = 	10
−6x1	 + 	3x2	 − 	15x3	 = 	−18
(x1, x2, x3) =

Let's solve the system of equations step by step. The system of linear equations is:

1. \( 2x_1 + x_2 - 3x_3 = 4 \)
2. \( 4x_1 + 2x_3 = 10 \)
3. \( -6x_1 + 3x_2 - 15x_3 = -18 \)

### Step 1: Solve the second equation for \( x_1 \)
The second equation is:

\[
4x_1 + 2x_3 = 10
\]

Divide through by 2 to simplify:

\[
2x_1 + x_3 = 5
\]

Now solve for \( x_1 \):

\[
x_1 = \frac{5 - x_3}{2}
\]

### Step 2: Substitute \( x_1 \) into the first and third equations

#### First equation:

The first equation is:

\[
2x_1 + x_2 - 3x_3 = 4
\]

Substitute \( x_1 = \frac{5 - x_3}{2} \) into this equation:

\[
2\left(\frac{5 - x_3}{2}\right) + x_2 - 3x_3 = 4
\]

Simplifying:

\[
(5 - x_3) + x_2 - 3x_3 = 4
\]

\[
5 - 4x_3 + x_2 = 4
\]

Now solve for \( x_2 \):

\[
x_2 = 4x_3 - 1
\]

#### Third equation:

The third equation is:

\[
-6x_1 + 3x_2 - 15x_3 = -18
\]

Substitute \( x_1 = \frac{5 - x_3}{2} \) and \( x_2 = 4x_3 - 1 \):

\[
-6\left(\frac{5 - x_3}{2}\right) + 3(4x_3 - 1) - 15x_3 = -18
\]

Simplifying:

\[
-3(5 - x_3) + 12x_3 - 3 - 15x_3 = -18
\]

\[
-15 + 3x_3 + 12x_3 - 3 - 15x_3 = -18
\]

\[
-18 = -18
\]

This is always true, which means the third equation is dependent on the others. So, there are infinitely many solutions.

### Step 3: Express \( x_1, x_2, x_3 \) in terms of a parameter

Let \( x_3 = t \) (where \( t \) is a free parameter). Then:

\[
x_1 = \frac{5 - t}{2}
\]
\[
x_2 = 4t - 1
\]
\[
x_3 = t
\]

Thus, the solution is:

\[
(x_1, x_2, x_3) = \left( \frac{5 - t}{2}, 4t - 1, t \right)
\]

### Final Answer:
\[
(x_1, x_2, x_3) = \left( \frac{5 - t}{2}, 4t - 1, t \right)
\]

Would you like more details or have any questions?

### Related Questions:
1. How can you check if a system of equations has no solution?
2. What are dependent and independent equations in a system?
3. How do you interpret parametric solutions geometrically?
4. Can you solve a system of equations using matrices?
5. How does the method of substitution compare to Gaussian elimination?

**Tip**: For large systems, matrix methods like Gaussian elimination or using inverses simplify the solving process.

Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.) 2x1 + x2 − 3x3 = 4; 4x1 + 2x3 = 10; −6x1 + 3x2 − 15x3 = −18

Explore how to solve a system of linear equations using substitution and express variables in terms of parameters. This detailed solution walks through each step, covering key algebraic methods and highlighting when a system has infinitely many solutions. Suitable for Grades 10-12.

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