Campos Mann
07/04/2024 · Primary School
Writing equations of lines parallel and perpendicular to a given line... Consider the line \( y=-\frac{7}{5} x-7 \) Find the equation of the line that is perpendicular to this line and passes through the point \( (7,-5) \). Find the equation of the line that is parallel to this line and passes through the point \( (7,-5) \). Note that the ALEKS graphing calculator mesting your answer.
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Step-by-step Solution
为了找到与给定直线 \( y = -\frac{7}{5}x - 7 \) 平行和垂直的直线，并经过点 \( (7, -5) \)，我们可以按照以下步骤进行：
### 步骤 1：找到垂直直线的方程
1. **确定给定直线的斜率**：
给定直线的方程是 \( y = -\frac{7}{5}x - 7 \)。其斜率 \( m_1 \) 为 \( -\frac{7}{5} \)。
2. **确定垂直直线的斜率**：
如果两条直线垂直，那么它们的斜率乘积为 -1。设垂直直线的斜率为 \( m_2 \)，则有：
\[
m_1 \times m_2 = -1
\]
\[
-\frac{7}{5} \times m_2 = -1
\]
解得：
\[
m_2 = \frac{5}{7}
\]
3. **使用点斜式方程**：
我们知道垂直直线经过点 \( (7, -5) \)，所以可以使用点斜式方程 \( y - y_1 = m(x - x_1) \) 来表示这条直线：
\[
y - (-5) = \frac{5}{7}(x - 7)
\]
简化方程：
\[
y + 5 = \frac{5}{7}x - 5
\]
\[
y = \frac{5}{7}x - 10
\]
### 步骤 2：找到平行直线的方程
1. **确定平行直线的斜率**：
平行直线与给定直线具有相同的斜率。因此，平行直线的斜率 \( m_3 \) 也是 \( -\frac{7}{5} \)。
2. **使用点斜式方程**：
我们知道平行直线经过点 \( (7, -5) \)，所以可以使用点斜式方程 \( y - y_1 = m(x - x_1) \) 来表示这条直线：
\[
y - (-5) = -\frac{7}{5}(x - 7)
\]
简化方程：
\[
y + 5 = -\frac{7}{5}x + 7
\]
\[
y = -\frac{7}{5}x + 2
\]
### 最终答案
- 垂直于给定直线并经过点 \( (7, -5) \) 的直线方程为：
\[
y = \frac{5}{7}x - 10
\]
- 平行于给定直线并经过点 \( (7, -5) \) 的直线方程为：
\[
y = -\frac{7}{5}x + 2
\]
Quick Answer
垂直直线方程：\( y = \frac{5}{7}x - 10 \)
平行直线方程：\( y = -\frac{7}{5}x + 2 \)
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