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:
,
The slope-intercept form of the line is .
And the standard form is found after some algebraic handling:
(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:
,
The slope-intercept form of the line is .
And the standard form is found after some algebraic handling:
(10)
The standard form of the line is .
4) If we know that and
, then the standard form of the family of lines is:
,
And the standard form is found after some algebraic handling:
,
(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:
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 .