Intersection of Two Lines in a Plane
The image contains a problem written in German, which translates to "Examine the intersection behavior of the given two lines in the plane" for the equations:
1. \(6x_1 + 3x_2 = 10\)
2. \(7x_1 - 2x_2 = -1\)
To find out if the lines intersect, we can use these two equations to solve for \(x_1\) and \(x_2\). This is a system of linear equations that we can solve using various methods such as substitution, elimination, or matrix methods. I will use the elimination method.
First, write the equations in standard form:
\(6x_1 + 3x_2 = 10\) (Equation 1)
\(7x_1 - 2x_2 = -1\) (Equation 2)
Our goal is to eliminate one of the variables. To do this, we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of \(x_2\) match:
\(12x_1 + 6x_2 = 20\) (Equation 3, multiplied by 2)
\(21x_1 - 6x_2 = -3\) (Equation 4, multiplied by 3)
Next, add the two equations together to eliminate \(x_2\):
\(12x_1 + 6x_2 + 21x_1 - 6x_2 = 20 - 3\)
\(33x_1 = 17\)
Solve for \(x_1\):
\(x_1 = \frac{17}{33}\)
\(x_1 = \frac{17}{33} * \frac{3}{3} = \frac{51}{99}\)
\(x_1 = \frac{1}{2}\)
Now that we have the value for \(x_1\), we can substitute it back into one of the original equations to find \(x_2\). Let's use Equation 1:
\(6(\frac{1}{2}) + 3x_2 = 10\)
\(3 + 3x_2 = 10\)
\(3x_2 = 10 - 3\)
\(3x_2 = 7\)
Solve for \(x_2\):
\(x_2 = \frac{7}{3}\)
So the solution to the system of equations, which represents the intersection point of the two lines, is \(x_1 = \frac{1}{2}\), \(x_2 = \frac{7}{3}\). The two lines intersect at this point.
