Question - Intersection of Two Lines in a Plane
Solution:
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.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com