perpendicular equation to yx4x+7 and (10,5)

It seems like you're asking for the equation of a line perpendicular to the line \( y = 4x + 7 \) that passes through the point \( (10, 5) \).

### Step 1: Find the slope of the given line
The given equation is in the slope-intercept form, \( y = mx + b \), where \( m \) is the slope. In this case, the slope \( m = 4 \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the slope of the perpendicular line is:
\[
m_{\text{perpendicular}} = -\frac{1}{4}
\]

### Step 3: Use the point-slope form of the equation
Now that we have the slope of the perpendicular line, we can use the point-slope form to find the equation of the line. The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Here, \( m = -\frac{1}{4} \) and \( (x_1, y_1) = (10, 5) \).

Substitute these values into the point-slope form:
\[
y - 5 = -\frac{1}{4}(x - 10)
\]

### Step 4: Simplify the equation
Now, simplify the equation to get it into slope-intercept form:
\[
y - 5 = -\frac{1}{4}(x - 10)
\]
Distribute the \( -\frac{1}{4} \):
\[
y - 5 = -\frac{1}{4}x + \frac{10}{4}
\]
\[
y - 5 = -\frac{1}{4}x + \frac{5}{2}
\]
Now, add 5 to both sides:
\[
y = -\frac{1}{4}x + \frac{5}{2} + 5
\]
Convert 5 into a fraction to combine with \( \frac{5}{2} \):
\[
y = -\frac{1}{4}x + \frac{5}{2} + \frac{10}{2}
\]
\[
y = -\frac{1}{4}x + \frac{15}{2}
\]

### Final Answer:
The equation of the line perpendicular to \( y = 4x + 7 \) that passes through \( (10, 5) \) is:
\[
y = -\frac{1}{4}x + \frac{15}{2}
\]

Would you like further clarification on any part of this? 

Here are 5 related questions you might find helpful:
1. How do you find the equation of a parallel line?
2. What is the significance of the negative reciprocal when finding perpendicular slopes?
3. How do you convert a point-slope form equation into slope-intercept form?
4. How do you graph lines given their equations?
5. How do you check if two lines are perpendicular?

**Tip:** Always remember that the product of the slopes of two perpendicular lines is \( -1 \).

perpendicular equation to y = 4x + 7 and (10,5)

Learn how to find the equation of a line perpendicular to y = 4x + 7 that passes through the point (10, 5). This problem covers key algebraic concepts such as slopes of perpendicular lines, point-slope form, and slope-intercept form. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation