Answer :
To find the equation of the line passing through the points [tex]\((3, -4)\)[/tex] and [tex]\((5, 1)\)[/tex], we need to determine the standard form of the line, which is [tex]\(Ax + By = C\)[/tex].
### Step 1: Calculate the slope of the line
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the given points [tex]\((3, -4)\)[/tex] and [tex]\((5, 1)\)[/tex]:
[tex]\[ m = \frac{1 - (-4)}{5 - 3} = \frac{1 + 4}{5 - 3} = \frac{5}{2} \][/tex]
### Step 2: Use the point-slope form to find the equation
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using one of our points, [tex]\((3, -4)\)[/tex], and the slope [tex]\(m = \frac{5}{2}\)[/tex]:
[tex]\[ y + 4 = \frac{5}{2} (x - 3) \][/tex]
### Step 3: Simplify to slope-intercept form
Expanding and simplifying the equation:
[tex]\[ y + 4 = \frac{5}{2}x - \frac{15}{2} \][/tex]
Subtracting 4 on both sides:
[tex]\[ y = \frac{5}{2}x - \frac{15}{2} - 4 \][/tex]
Combining the constants on the right-hand side:
[tex]\[ y = \frac{5}{2}x - \frac{15}{2} - \frac{8}{2} \][/tex]
[tex]\[ y = \frac{5}{2}x - \frac{23}{2} \][/tex]
### Step 4: Convert to standard form [tex]\(Ax + By = C\)[/tex]
We need to rearrange [tex]\(y = \frac{5}{2}x - \frac{23}{2}\)[/tex] to the standard form [tex]\(Ax + By = C\)[/tex].
First, clear the fractions by multiplying through by 2:
[tex]\[ 2y = 5x - 23 \][/tex]
Rearranging to standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ 5x - 2y = 23 \][/tex]
### Final Result
The equation of the line in standard form is [tex]\(5x - 2y = 23\)[/tex], which matches option A.
Therefore, the correct answer is:
A. [tex]\(5x - 2y = 23\)[/tex]
### Step 1: Calculate the slope of the line
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the given points [tex]\((3, -4)\)[/tex] and [tex]\((5, 1)\)[/tex]:
[tex]\[ m = \frac{1 - (-4)}{5 - 3} = \frac{1 + 4}{5 - 3} = \frac{5}{2} \][/tex]
### Step 2: Use the point-slope form to find the equation
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using one of our points, [tex]\((3, -4)\)[/tex], and the slope [tex]\(m = \frac{5}{2}\)[/tex]:
[tex]\[ y + 4 = \frac{5}{2} (x - 3) \][/tex]
### Step 3: Simplify to slope-intercept form
Expanding and simplifying the equation:
[tex]\[ y + 4 = \frac{5}{2}x - \frac{15}{2} \][/tex]
Subtracting 4 on both sides:
[tex]\[ y = \frac{5}{2}x - \frac{15}{2} - 4 \][/tex]
Combining the constants on the right-hand side:
[tex]\[ y = \frac{5}{2}x - \frac{15}{2} - \frac{8}{2} \][/tex]
[tex]\[ y = \frac{5}{2}x - \frac{23}{2} \][/tex]
### Step 4: Convert to standard form [tex]\(Ax + By = C\)[/tex]
We need to rearrange [tex]\(y = \frac{5}{2}x - \frac{23}{2}\)[/tex] to the standard form [tex]\(Ax + By = C\)[/tex].
First, clear the fractions by multiplying through by 2:
[tex]\[ 2y = 5x - 23 \][/tex]
Rearranging to standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ 5x - 2y = 23 \][/tex]
### Final Result
The equation of the line in standard form is [tex]\(5x - 2y = 23\)[/tex], which matches option A.
Therefore, the correct answer is:
A. [tex]\(5x - 2y = 23\)[/tex]
