Are 4x + 3y = 14 and 5x + 7y = 13:
parallel, intersect, or perpendicular
Get Eq. 1 into y = mx + b format:
4x + 3y = 14
3y = -4x + 14
Divide each side by 3
| 3y |
| 3 |
| = |
| -4x + 14 |
| 3 |
Simplifying, we have:
y = -1.3333333333333x + 4.6666666666667
Final Answer
Get Eq. 2 into y = mx + b format:
5x + 7y = 13
7y = -5x + 13
Divide each side by 7
| 7y |
| 7 |
| = |
| -5x + 13 |
| 7 |
Simplifying, we have:
y = -0.71428571428571x + 1.8571428571429
Final Answer
Set EQ1 = EQ2:
-1.3333333333333x + 4.6666666666667 = -0.71428571428571x + 1.8571428571429
-1.3333333333333x - -0.71428571428571x = 1.8571428571429 - 4.6666666666667
- 0.61904761904762x = -2.8095238095238
x = -2.8095238095238/ - 0.61904761904762
x = 4.5384615384615
Plug x into equation 1
y = -1.3333333333333 * (4.5384615384615) + 4.6666666666667
y = -6.0512820512821 + 4.6666666666667
y = -1.3846
Our intersection point = (4.5384615384615, -1.3846)
Calculate the product of the 2 slopes:
Slope 1 * Slope 2 = -1.3333333333333 * -0.71428571428571 = 0.95238095238095
Perpendicular Check
Since the product of the 2 slopes ≠ -1
The lines are not perpendicular
Line Relation Check
The 2 lines intersect at (4.5384615384615, -1.3846)
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) = | -0.71428571428571 --1.3333333333333 |
| 1 + -0.71428571428571 *-1.3333333333333 |
| tan(θ) = | 0.61904761904762 |
| 1 + 0.95238095238095 |
| tan(θ) = | 0.61904761904762 |
| 1.952380952381 |
tan(θ) = 0.31707317073171
θ = 17.5924
