To find the equation of a line parallel to the line given by [tex]\( y = 4x - 9 \)[/tex] and passing through the point [tex]\((-5, 3)\)[/tex], follow these steps:
1. Identify the slope of the given line:
- The given equation is [tex]\( y = 4x - 9 \)[/tex]. In slope-intercept form, [tex]\( y = mx + b \)[/tex], the coefficient of [tex]\( x \)[/tex] is the slope [tex]\( m \)[/tex].
- Therefore, the slope of the given line is [tex]\( m = 4 \)[/tex].
2. Recognize that parallel lines have the same slope:
- Since we want a line parallel to [tex]\( y = 4x - 9 \)[/tex], the new line will also have a slope [tex]\( m = 4 \)[/tex].
3. Use the point-slope form to find the equation of the new line:
- The point-slope form of the equation of a line 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.
- Given the point [tex]\((-5, 3)\)[/tex] and the slope [tex]\( 4 \)[/tex], substitute these values into the point-slope form:
[tex]\[
y - 3 = 4(x + 5)
\][/tex]
4. Simplify the equation to slope-intercept form:
- Expand the equation:
[tex]\[
y - 3 = 4x + 20
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 4x + 20 + 3
\][/tex]
[tex]\[
y = 4x + 23
\][/tex]
5. Write the final equation:
- The equation in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[
y = 4x + 23
\][/tex]
So, the equation of the line parallel to [tex]\( y = 4x - 9 \)[/tex] and passing through the point [tex]\((-5, 3)\)[/tex] is [tex]\( y = 4x + 23 \)[/tex].
