Answer:
1) The slope-intercept and standard forms are
and
, respectively.
2) The slope-intercept form of the line is
. The standard form of the line is
.
3) The slope-intercept form of the line is
. The standard form of the line is
.
4) The slope-intercept and standard forms of the family of lines are
and
,
, respectively.
5) The slope-intercept form of the line is
. The standard form of the line is
.
Step-by-step explanation:
From Analytical Geometry we know that the slope-intercept form of the line is represented by:
(1)
Where:
- Independent variable, dimensionless.
- Slope, dimensionless.
- y-Intercept, dimensionless.
- Dependent variable, dimensionless.
In addition, the standard form of the line is represented by the following model:
(2)
Where
,
are constant coefficients, dimensionless.
Now we process to resolve each problem:
1) If we know that
and
, then we know that the slope-intercept form of the line is:
(3)
And the standard form is found after some algebraic handling:
(4)
The slope-intercept and standard forms are
and
, respectively.
2) From Geometry we know that a line can be formed by two distinct points on a plane. If we know that
and
, then we construct the following system of linear equations:
(5)
(6)
The solution of the system is:
, ![b = -\frac{9}{2}](https://tex.z-dn.net/?f=b%20%3D%20-%5Cfrac%7B9%7D%7B2%7D)
The slope-intercept form of the line is
.
And the standard form is found after some algebraic handling:
![-\frac{5}{2}\cdot x +y = -\frac{9}{2}](https://tex.z-dn.net/?f=-%5Cfrac%7B5%7D%7B2%7D%5Ccdot%20x%20%2By%20%3D%20-%5Cfrac%7B9%7D%7B2%7D)
(7)
The standard form of the line is
.
3) From Geometry we know that a line can be formed by two distinct points on a plane. If we know that
and
, then we construct the following system of linear equations:
(8)
(9)
The solution of the system is:
, ![b = 5](https://tex.z-dn.net/?f=b%20%3D%205)
The slope-intercept form of the line is
.
And the standard form is found after some algebraic handling:
![-\frac{5}{2}\cdot x+y =5](https://tex.z-dn.net/?f=-%5Cfrac%7B5%7D%7B2%7D%5Ccdot%20x%2By%20%3D5)
(10)
The standard form of the line is
.
4) If we know that
and
, then the standard form of the family of lines is:
, ![\forall \,c \in \mathbb{R}](https://tex.z-dn.net/?f=%5Cforall%20%5C%2Cc%20%5Cin%20%5Cmathbb%7BR%7D)
And the standard form is found after some algebraic handling:
![-7\cdot y = -2\cdot x +c](https://tex.z-dn.net/?f=-7%5Ccdot%20y%20%3D%20-2%5Ccdot%20x%20%2Bc)
,
(11)
The slope-intercept and standard forms of the family of lines are
and
,
, respectively.
5) If we know that
and
, then the y-intercept of the line is:
![3\cdot 2 + b = -1](https://tex.z-dn.net/?f=3%5Ccdot%202%20%2B%20b%20%3D%20-1)
![b = -7](https://tex.z-dn.net/?f=b%20%3D%20-7)
Then, the slope-intercept form of the line is
.
And the standard form is found after some algebraic handling:
(12)
The standard form of the line is
.