Which Is Not An Equation Of The Line Going Through \[$(6,7)\$\] And \[$(2,-1)\$\]?A. \[$y = 2x - 5\$\] B. \[$y - 7 = 2(x - 6)\$\] C. \[$y + 1 = 2(x - 2)\$\] D. \[$y - 1 = 2(x + 2)\$\]
Which is not an equation of the line going through (6,7) and (2,-1)?
In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. Given two points on a line, we can determine the equation of the line using these forms. In this article, we will explore which of the given options is not an equation of the line going through the points (6,7) and (2,-1).
To find the equation of a line passing through two points, we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Calculating the Slope
The slope of the line passing through the points (6,7) and (2,-1) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (6,7) and (x2, y2) = (2,-1).
m = (-1 - 7) / (2 - 6) m = -8 / -4 m = 2
Equation of the Line
Now that we have the slope, we can use the point-slope form to find the equation of the line passing through the points (6,7) and (2,-1).
y - 7 = 2(x - 6)
This is the equation of the line in point-slope form.
Option Analysis
Let's analyze each of the given options to determine which one is not an equation of the line going through the points (6,7) and (2,-1).
Option A: y = 2x - 5
This option is in slope-intercept form, where the slope is 2 and the y-intercept is -5. However, we need to check if this equation passes through the points (6,7) and (2,-1).
Substituting the point (6,7) into the equation, we get:
7 = 2(6) - 5 7 = 12 - 5 7 = 7
This equation passes through the point (6,7).
Substituting the point (2,-1) into the equation, we get:
-1 = 2(2) - 5 -1 = 4 - 5 -1 = -1
This equation also passes through the point (2,-1).
Option B: y - 7 = 2(x - 6)
This option is in point-slope form, where the slope is 2 and the point (6,7) is used as the reference point. We can simplify this equation to:
y - 7 = 2(x - 6) y - 7 = 2x - 12 y = 2x - 5
This equation is the same as Option A, which we have already analyzed.
Option C: y + 1 = 2(x - 2)
This option is in point-slope form, where the slope is 2 and the point (2,-1) is used as the reference point. We can simplify this equation to:
y + 1 = 2(x - 2) y + 1 = 2x - 4 y = 2x - 5
This equation is the same as Option A, which we have already analyzed.
Option D: y - 1 = 2(x + 2)
This option is in point-slope form, where the slope is 2 and the point (2,-1) is used as the reference point. We can simplify this equation to:
y - 1 = 2(x + 2) y - 1 = 2x + 4 y = 2x + 5
This equation does not pass through the point (6,7), as we can see by substituting the point into the equation:
7 = 2(6) + 5 7 = 12 + 5 7 ≠17
Therefore, Option D is not an equation of the line going through the points (6,7) and (2,-1).
In our previous article, we explored which of the given options is not an equation of the line going through the points (6,7) and (2,-1). In this article, we will answer some frequently asked questions related to equations of lines, including the point-slope form, slope-intercept form, and standard form.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Q: How do I find the slope of a line passing through two points?
A: To find the slope of a line passing through two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points on the line.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Q: How do I convert a point-slope form equation to slope-intercept form?
A: To convert a point-slope form equation to slope-intercept form, you can simplify the equation by distributing the slope to the x-term and then combining like terms.
Q: What is the standard form of a linear equation?
A: The standard form of a linear equation is given by:
Ax + By = C
where A, B, and C are constants, and x and y are variables.
Q: How do I find the equation of a line passing through two points?
A: To find the equation of a line passing through two points, you can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Q: Can I use the slope-intercept form to find the equation of a line passing through two points?
A: Yes, you can use the slope-intercept form to find the equation of a line passing through two points. However, you will need to find the slope of the line first using the formula:
m = (y2 - y1) / (x2 - x1)
Q: How do I determine if an equation is in point-slope form, slope-intercept form, or standard form?
A: To determine if an equation is in point-slope form, slope-intercept form, or standard form, you can look for the following characteristics:
- Point-slope form: The equation will have the form y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope of the line.
- Slope-intercept form: The equation will have the form y = mx + b, where m is the slope of the line, and b is the y-intercept.
- Standard form: The equation will have the form Ax + By = C, where A, B, and C are constants, and x and y are variables.
In conclusion, understanding equations of lines is an essential concept in mathematics. By knowing the point-slope form, slope-intercept form, and standard form, you can find the equation of a line passing through two points, determine the slope and y-intercept of a line, and convert between different forms of a linear equation.