To find the equation of the line that passes through the point [tex]\((0,1)\)[/tex] and has a slope of [tex]\(\frac{4}{5}\)[/tex], we start with the slope-intercept form of the equation of a line, which is
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Given:
- Slope [tex]\( m = \frac{4}{5} \)[/tex]
- Y-intercept [tex]\( b = 1 \)[/tex]
Substitute these values into the slope-intercept form equation:
[tex]\[ y = \frac{4}{5}x + 1 \][/tex]
To convert this into the standard form [tex]\( Ax + By = C \)[/tex], we need to eliminate the fraction by multiplying every term by the denominator, which is 5, to get rid of the fraction:
[tex]\[ 5y = 4x + 5 \][/tex]
Now, we rearrange this equation to have all variables and constants on one side:
[tex]\[ -4x + 5y = 5 \][/tex]
Or, equivalently:
[tex]\[ 4y - 5x = 5 \][/tex]
So, the equation of the line in standard form is:
[tex]\[ 4y - 5x = 5 \][/tex]
Among the given choices, the correct one is:
A. [tex]\( 4y - 5x = 5 \)[/tex]
Therefore, the correct answer is:
1
