# Thread: Linear system, solutions and their geometric interpretation.

1. ## Linear system, solutions and their geometric interpretation.

Solve the following system of equations and give a geometrical interpretation of the result.
x + y + z = 6
2x + y − 3z = -5
4x − 5y + z = −3
Give a geometrical interpretation of the intersection of the planes with equations
x + y − 3 = 0
y + z + 5 = 0
x + z + 2 = 0
Determine a scalar equation for the plane that passes through the point (2, 0, −1) and is perpendicular to the line of intersection of the planes
2x + y – z + 5 = 0 and x + y + 2z + 7 = 0.

2. Originally Posted by ilovemymath
Solve the following system of equations and give a geometrical interpretation of the result.
x + y + z = 6
2x + y − 3z = -5
4x − 5y + z = −3
What methods have you tried?

3. Originally Posted by pickslides
What methods have you tried?
i havent tried anythign because i m taking online course and where my teacher doesnt teach my anything so though i would get my answer here....please help

4. Has your teacher mentioned what method is preferred? I.e. using substitutions or matrices?

5. substitutions

6. Here's the system.

$\displaystyle x + y + z = 6$ ...(1)
$\displaystyle 2x + y − 3z = -5$ ...(2)
$\displaystyle 4x - 5y + z = -3$ ...(3)

Now looking at (1)

$\displaystyle x + y + z = 6$

$\displaystyle z = 6-y - x$

Now we substitute where we see $\displaystyle z$ into both (2) and (3) giving

$\displaystyle 2x + y - 3(6-y - x) = -5$ ...(2)
$\displaystyle 4x - 5y + 6-y - x = -3$ ...(3)

Neatening this up gives

$\displaystyle 2x + y - 18+3y +3 x = -5$ ...(2)
$\displaystyle 4x - 5y + 6-y - x = -3$ ...(3)

Grouping like terms

$\displaystyle 5x + 4y = 13$ ...(2)
$\displaystyle 3x - 6y = -9$ ...(3)

Do you follow? Can you use this method again on (2) and (3) to finish it?

7. isn't it easier to do determinant of matrix ? so u can see how many there are (if any) solutions of the system and so it's much easier to understand the geometrical interpretation

how do the planes correspond between them, how do they (if they are) intersects, is it in one point or is it along the line or... and so on

8. I agree, but the OP may not have matrices in their skill set.