Answer:
Slope intercept form: [tex] \boxed{y = -4x + 21} [/tex]
Standard form: [tex] \boxed{4x + y = 21} [/tex].
Step-by-step explanation:
To write the equation of a line that passes through the point [tex](5,1)[/tex] and is parallel to the line defined by [tex]4x + y = 2[/tex], follow these steps:
Find the Slope of the Given Line
First, we need to rewrite the equation [tex]4x + y = 2[/tex] in slope-intercept form ([tex]y = mx + b[/tex]) to identify its slope ([tex]m[/tex]).
Rewrite [tex]4x + y = 2[/tex] in slope-intercept form:
[tex] y = -4x + 2 [/tex]
The slope ([tex]m[/tex]) of the given line [tex]4x + y = 2[/tex] is [tex]-4[/tex].
Determine the Slope of the Parallel Line
Parallel lines have the same slope. Therefore, the slope ([tex]m[/tex]) of our parallel line is also [tex]-4[/tex].
Use the Point-Slope Form to Find the Equation
We can now use the point-slope form of the equation of a line to find the equation of the parallel line passing through [tex](5,1)[/tex] with slope [tex]-4[/tex]:
[tex] y - y_1 = m(x - x_1) [/tex]
Substitute [tex]m = -4[/tex] and [tex](x_1, y_1) = (5,1)[/tex] into the point-slope form:
[tex] y - 1 = -4(x - 5) [/tex]
Convert to Slope-Intercept Form
Now, simplify and solve for [tex]y[/tex]:
[tex] y - 1 = -4x + 20 [/tex]
[tex] y = -4x + 20 + 1 [/tex]
[tex] y = -4x + 21 [/tex]
Therefore, the equation of the line in slope intercept form:
[tex] \boxed{y = -4x + 21} [/tex]
Convert to Standard Form
To convert to standard form ([tex]Ax + By = C[/tex]), rearrange the equation:
[tex] 4x + y = 21 [/tex]
Therefore, the equation of the line in standard form with smallest integer coefficients is:
[tex] \boxed{4x + y = 21} [/tex]
