Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find the equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex] that passes through a specific point and is parallel to a given line, follow these steps:
1. Identify the slope of the given line:
- The given line is [tex]\( y = 2x + 4 \)[/tex].
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
- For the given line, the slope [tex]\( m \)[/tex] is 2.
2. Use the fact that parallel lines have the same slope:
- Since the new line is parallel to the given line, it must have the same slope.
- Therefore, the slope [tex]\( m \)[/tex] of our new line is also 2.
3. Use the point-slope form of the equation of a line:
- The point-slope form 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.
- We are given the point [tex]\( (3, -2) \)[/tex].
- Thus, [tex]\( x_1 = 3 \)[/tex], [tex]\( y_1 = -2 \)[/tex], and the slope [tex]\( m = 2 \)[/tex].
4. Substitute the slope and the point into the point-slope form:
- Plugging in the values, we get:
[tex]\[ y - (-2) = 2(x - 3) \][/tex]
- Simplify the equation:
[tex]\[ y + 2 = 2(x - 3) \][/tex]
5. Convert the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
- Distribute the slope [tex]\( 2 \)[/tex] on the right-hand side:
[tex]\[ y + 2 = 2x - 6 \][/tex]
- Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[ y = 2x - 8 \][/tex]
So, the equation of the line in slope-intercept form that passes through the point [tex]\( (3, -2) \)[/tex] and is parallel to the line [tex]\( y = 2x + 4 \)[/tex] is:
[tex]\[ y = 2x - 8 \][/tex]
1. Identify the slope of the given line:
- The given line is [tex]\( y = 2x + 4 \)[/tex].
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
- For the given line, the slope [tex]\( m \)[/tex] is 2.
2. Use the fact that parallel lines have the same slope:
- Since the new line is parallel to the given line, it must have the same slope.
- Therefore, the slope [tex]\( m \)[/tex] of our new line is also 2.
3. Use the point-slope form of the equation of a line:
- The point-slope form 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.
- We are given the point [tex]\( (3, -2) \)[/tex].
- Thus, [tex]\( x_1 = 3 \)[/tex], [tex]\( y_1 = -2 \)[/tex], and the slope [tex]\( m = 2 \)[/tex].
4. Substitute the slope and the point into the point-slope form:
- Plugging in the values, we get:
[tex]\[ y - (-2) = 2(x - 3) \][/tex]
- Simplify the equation:
[tex]\[ y + 2 = 2(x - 3) \][/tex]
5. Convert the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
- Distribute the slope [tex]\( 2 \)[/tex] on the right-hand side:
[tex]\[ y + 2 = 2x - 6 \][/tex]
- Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[ y = 2x - 8 \][/tex]
So, the equation of the line in slope-intercept form that passes through the point [tex]\( (3, -2) \)[/tex] and is parallel to the line [tex]\( y = 2x + 4 \)[/tex] is:
[tex]\[ y = 2x - 8 \][/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.