
or
The equation of line t is y=
7
4
x+2. Perpendicular to line t is line u, which passes through the point (2,
–
2). What is the equation of line u?

To find the equation of the line \( u \), which is perpendicular to line \( t \) and passes through the point \( (2, -2) \), follow these steps:

1. **Determine the slope of line \( t \)**:
   The equation of line \( t \) is given as:
   \[
   y = \frac{7}{4}x + 2
   \]
   The slope of line \( t \) is \( \frac{7}{4} \).

2. **Find the slope of line \( u \)**:
   Since line \( u \) is perpendicular to line \( t \), its slope will be the negative reciprocal of the slope of \( t \). The negative reciprocal of \( \frac{7}{4} \) is:
   \[
   -\frac{4}{7}
   \]
   Thus, the slope of line \( u \) is \( -\frac{4}{7} \).

3. **Use the point-slope form to find the equation of line \( u \)**:
   The point-slope form of a line equation is:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( m \) is the slope, and \( (x_1, y_1) \) is a point on the line.

   Plug in the slope \( -\frac{4}{7} \) and the point \( (2, -2) \):
   \[
   y - (-2) = -\frac{4}{7}(x - 2)
   \]
   This simplifies to:
   \[
   y + 2 = -\frac{4}{7}x + \frac{8}{7}
   \]

4. **Convert to slope-intercept form \( y = mx + b \)**:
   Isolate \( y \):
   \[
   y = -\frac{4}{7}x + \frac{8}{7} - 2
   \]
   Convert \( -2 \) to a fraction with a denominator of 7:
   \[
   -2 = -\frac{14}{7}
   \]
   Now, combine the fractions:
   \[
   y = -\frac{4}{7}x + \frac{8}{7} - \frac{14}{7}
   \]
   \[
   y = -\frac{4}{7}x - \frac{6}{7}
   \]

The equation of line \( u \) is:
\[
y = -\frac{4}{7}x - \frac{6}{7}
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you find the equation of a line given two points?
2. What is the relationship between the slopes of parallel lines?
3. How can you determine if two lines are perpendicular from their equations?
4. How do you convert an equation from standard form to slope-intercept form?
5. What is the significance of the slope in the equation of a line?

**Tip:** When finding the equation of a line, always identify the slope first, as it determines the line's direction.

The equation of line t is y= 7/4x + 2. Perpendicular to line t is line u, which passes through the point (2, -2). What is the equation of line u?

Learn how to find the equation of a line perpendicular to y = 7/4x + 2 and passing through the point (2, -2). This problem covers essential algebraic concepts, such as the slope of perpendicular lines, point-slope form, and slope-intercept form. Suitable for students in Grades 8-10.

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