Question

ine t passes through (4, 5) and is perpendicular to the line shown on the coordinate grid.

a coordinate plane with a line passing through two points at 0 comma 3 and 5 comma 2

What is the equation of line t in standard form?

5x − y = 15
−5x − y = 15
x − 5y = −75
x + 5y = 75

Answer 1

To find the equation of line \( t \) that passes through the point \( (4, 5) \) and is perpendicular to the given line, we first need to determine the slope of the given line.

1. **Find the slope of the given line:**
The line passes through the points \( (0, 3) \) and \( (5, 2) \). We can calculate the slope \( m \) using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 3}{5 - 0} = \frac{-1}{5}
\]

So, the slope of the given line is \( -\frac{1}{5} \).

2. **Find the perpendicular slope:**
The slope of line \( t \), which is perpendicular to the given line, is the negative reciprocal of \( -\frac{1}{5} \):

\[
m_t = 5
\]

3. **Write the equation of line \( t \):**
Now, we can use the point-slope form of the line equation:

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

Using point \( (4, 5) \) and slope \( 5 \):

\[
y - 5 = 5(x - 4)
\]

Simplifying this:

\[
y - 5 = 5x - 20
\]
\[
y = 5x - 15
\]

4. **Convert to standard form:**
The standard form of a linear equation is \( Ax + By = C \). To convert \( y = 5x - 15 \):

\[
-5x + y = -15
\]

Multiplying by -1 to make the coefficients positive:

\[
5x - y = 15
\]

So, in standard form, the equation of line \( t \) is:

\[
5x - y = 15
\]

Thus, the correct answer is:

\[
\boxed{5x - y = 15}
\]