A coordinate plane with a line passing through the points (negative 4, negative 5) and (1, negative 1).Write an equation in standard form of the line that is graphed. Then find the x- and y-intercepts.The equation of the line in standard form is .The x-intercept is .The y-intercept is .

The equation of the line in standard form is
4x-5y=9?
5x-4y=9?
4x-5y=-1?

The x-intercept is
-9/5
1/5
9/4

The y-intercept is
-9/5
9/4
-5/4

1 answer

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 \(-\frac{9}{5}\).