To find the equation of the line in standard form that passes through the points \((-4, -5)\) and \((1, -1)\), we first calculate the slope of the line.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in the values of our points:
\[ m = \frac{-1 - (-5)}{1 - (-4)} = \frac{-1 + 5}{1 + 4} = \frac{4}{5} \]
Thus, the slope of the line is \(\frac{4}{5}\).
Next, we will use the point-slope form of the equation of a line which is:
\[ y - y_1 = m(x - x_1) \]
Using the point \((-4, -5)\) and the slope \(m = \frac{4}{5}\):
\[ y - (-5) = \frac{4}{5}(x - (-4)) \]
This simplifies to:
\[ y + 5 = \frac{4}{5}(x + 4) \]
Distributing the right side:
\[ y + 5 = \frac{4}{5}x + \frac{16}{5} \]
To isolate \(y\), we subtract 5 from both sides:
\[ y = \frac{4}{5}x + \frac{16}{5} - 5 \]
Since \(5\) can be written as \(\frac{25}{5}\):
\[ y = \frac{4}{5}x + \frac{16}{5} - \frac{25}{5} \]
This gives:
\[ y = \frac{4}{5}x - \frac{9}{5} \]
Now we will convert this into standard form \(Ax + By = C\). To eliminate the fraction, we can multiply the entire equation by 5:
\[ 5y = 4x - 9 \]
Rearranging gives:
\[ -4x + 5y = -9 \]
To write it in standard form \(Ax + By = C\) with \(A\) positive, we multiply through by \(-1\):
\[ 4x - 5y = 9 \]
So the equation of the line in standard form is:
4x - 5y = 9
Next, we find the x-intercept and y-intercept.
X-Intercept:
To find the x-intercept, set \(y = 0\):
\[ 4x - 5(0) = 9 \implies 4x = 9 \implies x = \frac{9}{4} \]
Y-Intercept:
To find the y-intercept, set \(x = 0\):
\[ 4(0) - 5y = 9 \implies -5y = 9 \implies y = -\frac{9}{5} \]
Summary:
The equation of the line in standard form is 4x - 5y = 9.
The x-intercept is \(\frac{9}{4}\).
The y-intercept is \(-\frac{9}{5}\).
So to finalize:
- The equation of the line in standard form is 4x - 5y = 9.
- The x-intercept is \(\frac{9}{4}\).
- The y-intercept is \(-\frac{9}{5}\).
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