What is the y intercept of the plane x-5y+7z=10

Answer

To find the y-intercept of the plane given by the equation \( x - 5y + 7z = 10 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation of the Plane**: The equation of the plane is given as:
\[ x - 5y + 7z = 10 \]

2. **Convert to Intercept Form**: The intercept form of a plane is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
where \( a \), \( b \), and \( c \) are the x, y, and z intercepts respectively.

3. **Rearranging the Equation**: We need to express the equation in the form of the intercept equation. First, we can rearrange the given equation:
\[ x - 5y + 7z = 10 \implies \frac{x}{10} - \frac{5y}{10} + \frac{7z}{10} = 1 \]
This can be simplified to:
\[ \frac{x}{10} - \frac{y}{2} + \frac{z}{\frac{10}{7}} = 1 \]

4. **Identify the Intercepts**: From the equation:
\[ \frac{x}{10} + \frac{y}{-2} + \frac{z}{\frac{10}{7}} = 1 \]
We can identify the intercepts:
- \( x \)-intercept \( (a) = 10 \)
- \( y \)-intercept \( (b) = -2 \)
- \( z \)-intercept \( (c) = \frac{10}{7} \)

5. **Conclusion**: Since we are interested in the y-intercept, we find that:
\[ \text{y-intercept} = -2 \]

### Final Answer:
The y-intercept of the plane \( x - 5y + 7z = 10 \) is \( -2 \).

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