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To solve this problem, we need to find the equation of a line that passes through the point \((-2, 4)\) and is perpendicular to the line given by the equation:

\[
2x + 3y = 21
\]

### Step 1: Find the slope of the given line.
We first need to rewrite the equation of the given line in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Starting with:
\[
2x + 3y = 21
\]

Solve for \(y\):

\[
3y = -2x + 21
\]
\[
y = -\frac{2}{3}x + 7
\]

Thus, the slope \(m\) of the given line is \(-\frac{2}{3}\).

### Step 2: Find the slope of the perpendicular line.
The slope of the line perpendicular to a given line is the negative reciprocal of the slope of the original line. The negative reciprocal of \(-\frac{2}{3}\) is \(\frac{3}{2}\).

So, the slope of the perpendicular line is \(m = \frac{3}{2}\).

### Step 3: Use the point-slope form to find the equation.
We know the slope of the perpendicular line is \(m = \frac{3}{2}\) and it passes through the point \((-2, 4)\). We can use the point-slope form of the equation of a line:

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

Substitute \(m = \frac{3}{2}\), \(x_1 = -2\), and \(y_1 = 4\):

\[
y - 4 = \frac{3}{2}(x - (-2))
\]
\[
y - 4 = \frac{3}{2}(x + 2)
\]
\[
y - 4 = \frac{3}{2}x + 3
\]
\[
y = \frac{3}{2}x + 7
\]

### Final Answer:
The equation of the line that passes through \((-2, 4)\) and is perpendicular to the line \(2x + 3y = 21\) is:

\[
y = \frac{3}{2}x + 7
\]

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Would you like further details or have any questions about this solution?

### Related Questions:
1. What is the formula for the negative reciprocal of a slope?
2. How can you convert an equation from standard form to slope-intercept form?
3. What is the point-slope form of a line, and how is it used?
4. How do you check if two lines are perpendicular?
5. What is the slope-intercept form of the equation of a line?

### Tip:
For any perpendicular line, always remember that their slopes multiply to \(-1\).

What is an equation of the line that passes through the point (-2, 4) and is perpendicular to the line 2x + 3y = 21?

Learn how to find the equation of a line that is perpendicular to the line 2x + 3y = 21 and passes through the point (-2, 4). This problem uses key algebraic concepts such as slopes of lines, perpendicularity, and the point-slope form. Suitable for students in Grades 9-11.

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