Find the distance between the lines with the equations
y = 3/2x+4 and −3x+2y = −1.
Round your answer to the nearest tenth.
Why did the line go to school?
To improve its line-teracy!
Now, let's find the distance between those lines. To do so, we can use the formula:
distance = |c1 - c2| / √(a^2 + b^2)
Where the equations are in the form ax + by + c = 0.
For the first equation, y = 3/2x + 4, we can rewrite it as 3x - 2y + 8 = 0.
Comparing this with the second equation, -3x + 2y + 1 = 0, we can see that a1 = 3, b1 = -2, and c1 = 8, while a2 = -3, b2 = 2, and c2 = 1.
Using the formula, the distance between the lines is:
distance = |c1 - c2| / √(a1^2 + b1^2)
Plugging in the values, we get:
distance = |8 - 1| / √((3)^2 + (-2)^2)
distance = 7 / √(9 + 4)
distance = 7 / √13
Now, rounding our answer to the nearest tenth, the distance between the lines is approximately 2.6 units.
To find the distance between the two lines, we need to find the shortest distance between any two points on the lines. We can do this by finding the perpendicular distance between the lines.
First, let's rewrite the second equation in slope-intercept form (y = mx + b).
-3x + 2y = -1
2y = 3x - 1
y = (3/2)x - 1/2
Now, we can compare the slopes of the two lines. The slope of the first line is 3/2, and the slope of the second line is also 3/2. Since the slopes are equal, the lines are parallel, and there is no intersection point.
Therefore, the shortest distance between the lines is 0.
So, the distance between the lines is 0 (rounded to the nearest tenth).
To find the distance between the lines, we can use the formula for the distance between a point and a line. In this case, we need to find the distance between a point on one line and the other line.
Step 1: Find a point on one line.
Let's choose the line y = 3/2x + 4. We can pick any x-value and find the corresponding y-value. For simplicity, let's choose x = 0.
When x = 0, y = 3/2(0) + 4 = 4.
So, we have a point (0, 4) on the line y = 3/2x + 4.
Step 2: Find the perpendicular distance from the point to the other line.
The perpendicular distance between a point (x1, y1) and a line Ax + By + C = 0 is given by the formula:
Distance = |Ax1 + By1 + C| / √(A^2 + B^2)
In this case, we have the line -3x + 2y = -1. Rewriting it in the standard form Ax + By + C = 0, we have -3x + 2y + 1 = 0.
Comparing with the formula, we have:
A = -3, B = 2, C = 1.
Substituting the values of (x1, y1) = (0, 4) and (A, B, C) into the formula, we get:
Distance = |-3(0) + 2(4) + 1| / √((-3)^2 + 2^2)
Distance = |0 + 8 + 1| / √(9 + 4)
Distance = 9 / √13
Step 3: Round the answer to the nearest tenth.
Using a calculator, we find that Distance ≈ 2.4 (rounded to the nearest tenth).
Therefore, the distance between the lines y = 3/2x + 4 and −3x + 2y = −1 is approximately 2.4 units.
y = 3/2x+4
2y = 3x + 8
3x - 2y + 8 = 0
-3x + 2y = -1
3x - 2y - 1 = 0 , so they are indeed parallel
pick any point on the first line,
e.g. let x = 2, then y = 7
so (2,7) lies on the first
distance from (2,7) to 3x - 2y - 1 = 0 is
|2(3) - 2(7) -1|/√(3^2 + (-2)^2)
= 9/√13
= appr 2.5