https://file.aiquickdraw.com/mathbot/file/1730281163835sn511fma.jpg

Let's solve this task step-by-step.

### Task 2.25

The problem states:
The intersection of planes \( 3x + 2y - z + 1 = 0 \) and \( x + 3y - 2z + 2 = 0 \) is a line.

1. **Part (a):** Find the parametric equation of the line.
2. **Part (b):** Find the implicit equation of a plane that contains the line from (a) and is perpendicular to the plane \( z = 0 \).

### Solution

#### Part (a): Parametric Equation of the Line

To find the line of intersection between the two planes, we solve the system of equations:
\[
3x + 2y - z + 1 = 0
\]
\[
x + 3y - 2z + 2 = 0
\]

**Step 1:** Eliminate \( z \) by manipulating the equations.

1. Multiply the first equation by 2:
   \[
   6x + 4y - 2z + 2 = 0
   \]
2. Add this to the second equation:
   \[
   (6x + 4y - 2z + 2) + (x + 3y - 2z + 2) = 0
   \]
   Simplifying, we get:
   \[
   7x + 7y + 4 = 0 \Rightarrow x + y = -\frac{4}{7}
   \]

**Step 2:** Solve for \( x \) in terms of \( y \) and \( z \).

Let \( t \) be a parameter. Choose \( x = t \). Then,
\[
y = -5t
\]
and
\[


Zadatak 2.25. Presjek ravnina 3x + 2y − z + 1 = 0 i x + 3y − 2z + 2 = 0 je pravac. a) Odredite parametarsku jednadžbu tog pravca. b) Odredite implicitnu jednadžbu ravnine koja sadrži pravac iz a) dijela zadatka i okomita je na ravninu z = 0.

Solve the intersection problem between planes 3x + 2y − z + 1 = 0 and x + 3y − 2z + 2 = 0. Learn how to find the parametric equation of the intersecting line and the implicit equation of a plane containing the line, perpendicular to z = 0. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation