Answer :
To determine which equation represents the line that is perpendicular to the given line [tex]\(4x - 5y = 5\)[/tex] and passes through the point [tex]\((5,3)\)[/tex], follow these steps:
### Step 1: Find the slope of the given line
To find the slope of the given line [tex]\(4x - 5y = 5\)[/tex], we rewrite it in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Starting with:
[tex]\[4x - 5y = 5\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -5y = -4x + 5 \][/tex]
[tex]\[ y = \frac{4}{5}x - 1 \][/tex]
So, the slope [tex]\(m_\text{given}\)[/tex] of the given line is [tex]\(\frac{4}{5}\)[/tex].
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line with slope [tex]\(m\)[/tex] is the negative reciprocal of [tex]\(m\)[/tex].
The slope of the given line is [tex]\(\frac{4}{5}\)[/tex], so the slope [tex]\(m_\perp\)[/tex] of the perpendicular line is:
[tex]\[ m_\perp = -\frac{1}{m_\text{given}} = -\frac{1}{\left(\frac{4}{5}\right)} = -\frac{5}{4} \][/tex]
### Step 3: Use the point-slope form to write the equation of the perpendicular line
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given point [tex]\((5,3)\)[/tex] and slope [tex]\(m_\perp = -\frac{5}{4}\)[/tex], we substitute these values into the point-slope form:
[tex]\[ y - 3 = -\frac{5}{4}(x - 5) \][/tex]
### Step 4: Simplify to get the equation in standard form
First, distribute [tex]\( -\frac{5}{4} \)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}x + \frac{25}{4} \][/tex]
Add 3 to both sides:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + 3 \][/tex]
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \][/tex]
[tex]\[ y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
Rewrite this in standard form [tex]\(Ax + By = C\)[/tex], multiply through by 4 to clear fractions:
[tex]\[ 4y = -5x + 37 \][/tex]
[tex]\[ 5x + 4y = 37 \][/tex]
So, the equation of the line that is perpendicular to the given line [tex]\(4x - 5y = 5\)[/tex] and passes through the point [tex]\((5, 3)\)[/tex] is:
[tex]\[ 5x + 4y = 37 \][/tex]
Thus, the correct answer is:
[tex]\[ 5x + 4y = 37 \][/tex]
### Step 1: Find the slope of the given line
To find the slope of the given line [tex]\(4x - 5y = 5\)[/tex], we rewrite it in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Starting with:
[tex]\[4x - 5y = 5\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -5y = -4x + 5 \][/tex]
[tex]\[ y = \frac{4}{5}x - 1 \][/tex]
So, the slope [tex]\(m_\text{given}\)[/tex] of the given line is [tex]\(\frac{4}{5}\)[/tex].
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line with slope [tex]\(m\)[/tex] is the negative reciprocal of [tex]\(m\)[/tex].
The slope of the given line is [tex]\(\frac{4}{5}\)[/tex], so the slope [tex]\(m_\perp\)[/tex] of the perpendicular line is:
[tex]\[ m_\perp = -\frac{1}{m_\text{given}} = -\frac{1}{\left(\frac{4}{5}\right)} = -\frac{5}{4} \][/tex]
### Step 3: Use the point-slope form to write the equation of the perpendicular line
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given point [tex]\((5,3)\)[/tex] and slope [tex]\(m_\perp = -\frac{5}{4}\)[/tex], we substitute these values into the point-slope form:
[tex]\[ y - 3 = -\frac{5}{4}(x - 5) \][/tex]
### Step 4: Simplify to get the equation in standard form
First, distribute [tex]\( -\frac{5}{4} \)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}x + \frac{25}{4} \][/tex]
Add 3 to both sides:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + 3 \][/tex]
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \][/tex]
[tex]\[ y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
Rewrite this in standard form [tex]\(Ax + By = C\)[/tex], multiply through by 4 to clear fractions:
[tex]\[ 4y = -5x + 37 \][/tex]
[tex]\[ 5x + 4y = 37 \][/tex]
So, the equation of the line that is perpendicular to the given line [tex]\(4x - 5y = 5\)[/tex] and passes through the point [tex]\((5, 3)\)[/tex] is:
[tex]\[ 5x + 4y = 37 \][/tex]
Thus, the correct answer is:
[tex]\[ 5x + 4y = 37 \][/tex]
