Problem 78 - Calculus: One and Several Variables

Question

Find the point where the lines intersect and determine the angle between the lines. $$l_{1}: 4 x-y+2=0, \quad l_{2}: 19 x+y=0$$.

Step-by-Step Solution

Answer

The point of intersection for the lines is \((-2/23, 38/23)\) and the angle between them is approximately \(-17.01\) degrees or \(-0.2967\) radians.

Step 1: Solve the system of linear equations

For the intersecting point of two lines, the x and y values will satisfy both equations. So to find the intersection point (x, y), we solve the system of equations:\n\(4x - y + 2 = 0\)\nand\n\(19x + y = 0\)\nAdding these two equations, we get:\n\(23x + 2 = 0\)\nFrom which we get\n\(x = -2/23\)\nThen substitute x into either of original equations to solve for y. If we substitute into the second equation we get: \n\(19*(-2/23) + y = 0\)\nThis simplifies to \(-38/23 + y = 0\), then \(y = 38/23\)

Step 2: Find the slopes

The slope (m) of a line given by the equation \(ax + by + c = 0\) is \(-a/b\). So, for line \(l1: 4x - y + 2 = 0\), the slope \(m1\) is \(-4/-1 = 4\). For line \(l2: 19x + y = 0\), the slope \(m2\) is \(-19/1 = -19\)

Step 3: Calculate the angle

The formula for calculating the angle between two lines is \(\tan(\theta) = \frac{m1 - m2}{1 + m1*m2}\)\nSubstitute the slopes of lines \(l1\) and \(l2\) to get:\n\(\tan(\theta) = \frac{4 - (-19)}{1 + 4*(-19)} = 23/(-75)\)\nThe angle \(\theta\) then is \(\arctan(23/-75)\) which approximately equals to \(-0.2967\) radians or \(-17.01\) degrees

Key Concepts

System of Linear Equations

In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. In this exercise, we are dealing with two equations, representing two lines, and we want to find the point where these lines intersect.
For lines to intersect, the values of variables, usually denoted by x and y, will satisfy both equations simultaneously. Here, the two equations given are:

• Equation 1: \(4x - y + 2 = 0\)
• Equation 2: \(19x + y = 0\)

To solve this system, we can use methods like substitution or elimination. In this solution, we added the equations to eliminate y, finding the value of x. After that, we substituted x back into one of the equations to find y. This process revealed the intersection point, where both lines share the same x and y values, in this case, \((-2/23, 38/23)\).

Slope of a Line

The slope of a line is a number that describes both the direction and the steepness of the line. It can be calculated if the line is given in the standard form equation, \(ax + by + c = 0\), as \(-\frac{a}{b}\).
In our original exercise, we calculated:

• For line \(l_1 : 4x - y + 2 = 0\), the slope \(m_1\) is \(\frac{4}{1} = 4\).
• For line \(l_2 : 19x + y = 0\), the slope \(m_2\) is \(-19\).

The slope tells us about the orientation of the line. A positive slope means the line increases from left to right, while a negative slope means it decreases. Here, line \(l_1\) increases steeply, while line \(l_2\) decreases very steeply.

Angle Between Lines

The angle between two lines can be determined using their slopes. The formula to find this angle, \(\theta\), is given by:\[\tan(\theta) = \frac{m_1 - m_2}{1 + m_1 \cdot m_2}\]This formula helps to summarize how two different slopes interact to form an angle.
Substituting our calculated slopes, \(m_1 = 4\) and \(m_2 = -19\), into the formula, we have:

• \(\tan(\theta) = \frac{4 - (-19)}{1 + 4 \times (-19)} = \frac{23}{-75}\)

To find \(\theta\), we use the inverse tangent function (\(\arctan\)), which reveals the angle in radians or degrees.
Specifically for our case, \(\arctan\left(\frac{23}{-75}\right)\), gives an angle of approximately \(-0.2967\) radians or \(-17.01\) degrees. The negative sign indicates the angle direction.