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We are given the equation of line \( t \) in point-slope form:

\[
y - 1 = -\frac{2}{5}(x + 3)
\]

### Step 1: Determine the slope of line \( t \)

The equation is in point-slope form:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope. From the equation of line \( t \), we can see that the slope \( m_t \) is:

\[
m_t = -\frac{2}{5}
\]

### Step 2: Find the slope of line \( u \) (perpendicular to \( t \))

If two lines are perpendicular, their slopes are negative reciprocals of each other. So, the slope \( m_u \) of line \( u \) will be the negative reciprocal of \( -\frac{2}{5} \):

\[
m_u = \frac{5}{2}
\]

### Step 3: Use the point \( (1, 2) \) to find the equation of line \( u \)

We are given that line \( u \) passes through the point \( (1, 2) \) and has a slope of \( \frac{5}{2} \). Using the point-slope form of a line's equation again:

\[
y - y_1 = m(x - x_1)
\]

Substitute \( (x_1, y_1) = (1, 2) \) and \( m = \frac{5}{2} \):

\[
y - 2 = \frac{5}{2}(x - 1)
\]

### Step 4: Simplify to slope-intercept form

Now, we simplify this equation to slope-intercept form \( y = mx + b \):

\[
y - 2 = \frac{5}{2}(x - 1)
\]

Distribute the \( \frac{5}{2} \):

\[
y - 2 = \frac{5}{2}x - \frac{5}{2}
\]

Add 2 to both sides:

\[
y = \frac{5}{2}x - \frac{5}{2} + 2
\]

Convert 2 to a fraction with denominator 2:

\[
y = \frac{5}{2}x - \frac{5}{2} + \frac{4}{2}
\]

Simplify the constants:

\[
y = \frac{5}{2}x - \frac{1}{2}
\]

### Final Answer:

The equation of line \( u \) in slope-intercept form is:

\[
y = \frac{5}{2}x - \frac{1}{2}
\]

Would you like more details on any of the steps? Here are 5 related questions for further exploration:

1. How do you find the equation of a line parallel to a given line?
2. What is the significance of negative reciprocal slopes in perpendicular lines?
3. How can you convert a point-slope equation to standard form?
4. Why does the slope-intercept form of a line allow quick identification of the slope and y-intercept?
5. How would the equation change if line \( u \) passed through a different point?

**Tip**: Always double-check that you have simplified your equation to its cleanest form for clarity in results.

Line t has an equation of y - 1 = -(2/5)(x + 3). Perpendicular to line t is line u, which passes through the point (1, 2). What is the equation of line u?

Learn how to find the equation of a line perpendicular to y - 1 = -(2/5)(x + 3), passing through the point (1, 2). This algebra problem covers key concepts such as point-slope form and slope-intercept form. Perfect for students in Grades 8-10.

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