Use intercepts to help sketch the plane: $3x + 2y + z = 6$

• $x$-intercept: $x = \boxed \,$
• $y$-intercept: $y = \boxed \,$
• $z$-intercept: $z = \boxed \,$

Neetesh Kumar | December 16, 2024

Calculus Homework Help

This is the solution to Math 1C
Assignment: 12.5 Question Number 18
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Step-by-step solution:

To find the intercepts of the plane, we set two of the variables to zero and solve for the third variable.

Step 1: Find the x-intercept

The x-intercept occurs when $y = 0$ and $z = 0$.

Substitute $y = 0$ and $z = 0$ into the plane equation:

$3x + 2(0) + 0 = 6$
$3x = 6$
$x = 2$

So, the x-intercept is $x = 2$.

Step 2: Find the y-intercept

The y-intercept occurs when $x = 0$ and $z = 0$.

Substitute $x = 0$ and $z = 0$ into the plane equation:

$3(0) + 2y + 0 = 6$
$2y = 6$
$y = 3$

So, the y-intercept is $y = 3$.

Step 3: Find the z-intercept

The z-intercept occurs when $x = 0$ and $y = 0$.

Substitute $x = 0$ and $y = 0$ into the plane equation:

$3(0) + 2(0) + z = 6$
$z = 6$

So, the z-intercept is $z = 6$.

Final Answer:

• $x$-intercept: $x = \boxed{2}$

• $y$-intercept: $y = \boxed{3}$

• $z$-intercept: $z = \boxed{6}$

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