To find the equation of the line in slope-intercept form ([tex]\(y = mx + b\)[/tex]) that passes through the point [tex]\((6, 1)\)[/tex] and is parallel to the line [tex]\(4x - y = 3\)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(4x - y = 3\)[/tex]. To find its slope, we need to rewrite this line in slope-intercept form ([tex]\(y = mx + b\)[/tex]).
Starting from the given equation:
[tex]\[
4x - y = 3
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
-y = -4x + 3
\][/tex]
[tex]\[
y = 4x - 3
\][/tex]
From the equation [tex]\(y = 4x - 3\)[/tex], we can see that the slope ([tex]\(m\)[/tex]) of the given line is 4.
2. Determine the slope of the new line:
Since the required line is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also 4.
3. Use the point-slope form to write the equation:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope. Substituting the point [tex]\((6, 1)\)[/tex] and the slope 4:
[tex]\[
y - 1 = 4(x - 6)
\][/tex]
4. Simplify to get the slope-intercept form:
Distribute the slope on the right side:
[tex]\[
y - 1 = 4x - 24
\][/tex]
Add 1 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = 4x - 24 + 1
\][/tex]
[tex]\[
y = 4x - 23
\][/tex]
Therefore, the equation of the line in slope-intercept form that passes through the point [tex]\((6, 1)\)[/tex] and is parallel to the line [tex]\(4x - y = 3\)[/tex] is:
[tex]\[
y = 4x - 23
\][/tex]
