Are 3x + 9y = 19 and 8x + 4y = 10:
parallel, intersect, or perpendicular
Get Eq. 1 into y = mx + b format:
3x + 9y = 19
9y = -3x + 19
Divide each side by 9
| 9y |
| 9 |
| = |
| -3x + 19 |
| 9 |
Simplifying, we have:
y = -0.33333333333333x + 2.1111111111111
Final Answer
Get Eq. 2 into y = mx + b format:
8x + 4y = 10
4y = -8x + 10
Divide each side by 4
| 4y |
| 4 |
| = |
| -8x + 10 |
| 4 |
Simplifying, we have:
y = -2x + 2.5
Final Answer
Set EQ1 = EQ2:
-0.33333333333333x + 2.1111111111111 = -2x + 2.5
-0.33333333333333x - -2x = 2.5 - 2.1111111111111
+ 1.6666666666667x = 0.38888888888889
x = 0.38888888888889/ + 1.6666666666667
x = 0.23333333333333
Plug x into equation 1
y = -0.33333333333333 * (0.23333333333333) + 2.1111111111111
y = -0.077777777777778 + 2.1111111111111
y = 2.0333
Our intersection point = (0.23333333333333, 2.0333)
Calculate the product of the 2 slopes:
Slope 1 * Slope 2 = -0.33333333333333 * -2 = 0.66666666666667
Perpendicular Check
Since the product of the 2 slopes ≠ -1
The lines are not perpendicular
Line Relation Check
The 2 lines intersect at (0.23333333333333, 2.0333)
Check if equations are Independent:
Since the slopes are different
The systems are independent
Check if equations are Dependent:
To be dependent
the slopes and y-intercept must be the same.
This is not the case
Check if equations are Inconsistent:
To be inconsistent
The slopes must be the same
Ty-intercepts must different.
This is not the case
Calculate the angle θ
θ is formed by the two lines
| tan(θ) = | m2 - m1 |
| 1 + m2m1 |
| tan(θ) = | -2 --0.33333333333333 |
| 1 + -2 *-0.33333333333333 |
| tan(θ) = | -1.6666666666667 |
| 1 + 0.66666666666667 |
| tan(θ) = | -1.6666666666667 |
| 1.6666666666667 |
tan(θ) = -1
θ = -45
Common Core State Standards In This Lesson
What is the Answer?
How does the 2 Lines Intersection Calculator work?
* Are the lines parallel?
* Are the lines perpendicular
* Do the lines intersect at some point, and if so, which point?
* Is the system of equations dependent, independent, or inconsistent
This calculator has 2 inputs.
What 3 formulas are used for the 2 Lines Intersection Calculator?
Lines are perpendicular when they have slopes which are negative reciprocals.
For a line y = mx + b, m is the slope
What 6 concepts are covered in the 2 Lines Intersection Calculator?
- 2 lines intersection
- intersect
- line equation
- parallel
- perpendicular
- two lines which intersect to form a right angle
- slope
- Change in y over change in x
