Question

|3х + 2y = 2, 2. Система уравненийимеет единственное решение.|3х - 2y = 1Тогда графики уравнений системы:а) совпадают; б) параллельны; в) пересекаются.​

Answer 1

3x+2y=2
3x-2y=1

Домножаем 2 уравнение на -1

3x+2y=2
-3x+2y=-1

4y=1
-3x+2y=-1

y=1/4
-3x+1/2=-1

y=1/4
-6x+1=-1

y=1/4
-6x=-2

y=1/4
x=1/3
короче в) графики пересекаются

Answer 2

To determine the relative positions of the graphs of the two equations, let's analyze the system of equations:

1. 3x + 2y = 2
2. 3x - 2y = 1

First, let's find the solution to the system of equations. We can use the method of elimination:

Add the two equations together:

(3x + 2y) + (3x - 2y) = 2 + 1

Simplifying the equation:

6x = 3

Now, solve for x:

x = 3/6 x = 1/2

Now that we have the value of x, we can find the value of y by substituting x into one of the equations. Let's use the first equation:

3(1/2) + 2y = 2

3/2 + 2y = 2

Subtract 3/2 from both sides:

2y = 2 - 3/2 2y = 4/2 - 3/2 2y = 1/2

Now, solve for y:

y = (1/2) / 2 y = 1/4

So, the solution to the system is x = 1/2 and y = 1/4. Now, let's analyze the options for the relative positions of the graphs:

a) The graphs sоvpadают (coincide): This would mean that the two equations represent the same line, and they have infinitely many solutions. However, since we found that the system has a unique solution, this option can be eliminated.

b) The graphs parаllеlny (are parallel): If the graphs are parallel, the two equations represent lines that have the same slope but different y-intercepts. In this case, the slopes of both equations are equal to 3, which is not the case since they have opposite signs.

c) The graphs pеrеsекаются (intersect): This is the correct option. The system has a unique solution, which means that the graphs of the two equations intersect at a single point. This point is (1/2, 1/4), which is the solution we found earlier.

Therefore, the correct answer is c) The graphs intersect.