Question

list the 3 intercepts for each problem

1. $6x - 3y + 2z = 6$
2. $-x + y - 5z = 5$
3. $2x - 4y - 2z = 4$
4. $3x + 2y - 2z = -6$

Explanation:

Problem 3: \( 6x - 3y + 2z = 6 \)

Step 1: Find x-intercept (set \( y = 0, z = 0 \))

Substitute \( y = 0 \) and \( z = 0 \) into the equation:
\( 6x - 3(0) + 2(0) = 6 \)
\( 6x = 6 \)
\( x = \frac{6}{6} = 1 \)
x-intercept: \( (1, 0, 0) \)

Step 2: Find y-intercept (set \( x = 0, z = 0 \))

Substitute \( x = 0 \) and \( z = 0 \) into the equation:
\( 6(0) - 3y + 2(0) = 6 \)
\( -3y = 6 \)
\( y = \frac{6}{-3} = -2 \)
y-intercept: \( (0, -2, 0) \)

Step 3: Find z-intercept (set \( x = 0, y = 0 \))

Substitute \( x = 0 \) and \( y = 0 \) into the equation:
\( 6(0) - 3(0) + 2z = 6 \)
\( 2z = 6 \)
\( z = \frac{6}{2} = 3 \)
z-intercept: \( (0, 0, 3) \)

Step 1: Find x-intercept (set \( y = 0, z = 0 \))

Substitute \( y = 0 \) and \( z = 0 \) into the equation:
\( 2x - 4(0) - 2(0) = 4 \)
\( 2x = 4 \)
\( x = \frac{4}{2} = 2 \)
x-intercept: \( (2, 0, 0) \)

Step 2: Find y-intercept (set \( x = 0, z = 0 \))

Substitute \( x = 0 \) and \( z = 0 \) into the equation:
\( 2(0) - 4y - 2(0) = 4 \)
\( -4y = 4 \)
\( y = \frac{4}{-4} = -1 \)
y-intercept: \( (0, -1, 0) \)

Step 3: Find z-intercept (set \( x = 0, y = 0 \))

Substitute \( x = 0 \) and \( y = 0 \) into the equation:
\( 2(0) - 4(0) - 2z = 4 \)
\( -2z = 4 \)
\( z = \frac{4}{-2} = -2 \)
z-intercept: \( (0, 0, -2) \)

Step 1: Find x-intercept (set \( y = 0, z = 0 \))

Substitute \( y = 0 \) and \( z = 0 \) into the equation:
\( -x + 0 - 5(0) = 5 \)
\( -x = 5 \)
\( x = -5 \)
x-intercept: \( (-5, 0, 0) \)

Step 2: Find y-intercept (set \( x = 0, z = 0 \))

Substitute \( x = 0 \) and \( z = 0 \) into the equation:
\( -0 + y - 5(0) = 5 \)
\( y = 5 \)
y-intercept: \( (0, 5, 0) \)

Step 3: Find z-intercept (set \( x = 0, y = 0 \))

Substitute \( x = 0 \) and \( y = 0 \) into the equation:
\( -0 + 0 - 5z = 5 \)
\( -5z = 5 \)
\( z = \frac{5}{-5} = -1 \)
z-intercept: \( (0, 0, -1) \)

Answer:

x-intercept: \( (1, 0, 0) \), y-intercept: \( (0, -2, 0) \), z-intercept: \( (0, 0, 3) \)

Problem 4: \( 2x - 4y - 2z = 4 \)