Answer :
To find the equation of a line that is parallel to [tex]\( y = \frac{1}{2}x - 4 \)[/tex] and passes through the point [tex]\( (3, -2) \)[/tex], you'll follow these steps:
1. Identify the Slope:
The equation given is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, [tex]\( m = \frac{1}{2} \)[/tex].
2. Understand Parallel Lines:
Lines that are parallel have the same slope. Therefore, the slope of our new line will also be [tex]\( \frac{1}{2} \)[/tex].
3. Use the Point-Slope Form:
Since the new line passes through the point [tex]\( (3, -2) \)[/tex], we use the point-slope form of a line, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the point [tex]\( (3, -2) \)[/tex] and the slope [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
y - (-2) = \frac{1}{2}(x - 3)
\][/tex]
Simplify this to:
[tex]\[
y + 2 = \frac{1}{2}x - \frac{3}{2}
\][/tex]
4. Convert to Slope-Intercept Form:
Isolate [tex]\( y \)[/tex] to express the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - 2
\][/tex]
Calculate the value:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - \frac{4}{2}
\][/tex]
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]
5. Final Equation:
The equation of the line parallel to [tex]\( y = \frac{1}{2}x - 4 \)[/tex] and passing through [tex]\( (3, -2) \)[/tex] is:
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]
The equation of the line that satisfies the given conditions is [tex]\( y = \frac{1}{2}x - \frac{7}{2} \)[/tex].
1. Identify the Slope:
The equation given is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, [tex]\( m = \frac{1}{2} \)[/tex].
2. Understand Parallel Lines:
Lines that are parallel have the same slope. Therefore, the slope of our new line will also be [tex]\( \frac{1}{2} \)[/tex].
3. Use the Point-Slope Form:
Since the new line passes through the point [tex]\( (3, -2) \)[/tex], we use the point-slope form of a line, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the point [tex]\( (3, -2) \)[/tex] and the slope [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
y - (-2) = \frac{1}{2}(x - 3)
\][/tex]
Simplify this to:
[tex]\[
y + 2 = \frac{1}{2}x - \frac{3}{2}
\][/tex]
4. Convert to Slope-Intercept Form:
Isolate [tex]\( y \)[/tex] to express the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - 2
\][/tex]
Calculate the value:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - \frac{4}{2}
\][/tex]
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]
5. Final Equation:
The equation of the line parallel to [tex]\( y = \frac{1}{2}x - 4 \)[/tex] and passing through [tex]\( (3, -2) \)[/tex] is:
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]
The equation of the line that satisfies the given conditions is [tex]\( y = \frac{1}{2}x - \frac{7}{2} \)[/tex].
