Which graph represents the solution to the given system? –2x + 5y = –10 and –3x + 5y = –20
To determine which graph represents the solution to the given system, we need to solve the system of equations first.
Let's solve the given system:
–2x + 5y = –10 ----(equation 1)
–3x + 5y = –20 ----(equation 2)
We can solve this system using the method of elimination:
Multiply equation 1 by 3:
3(-2x + 5y) = 3(-10)
-6x + 15y = -30 ----(equation 3)
Next, subtract equation 3 from equation 2:
(-3x + 5y) - (-6x + 15y) = (-20) - (-30)
-3x + 5y + 6x - 15y = -20 + 30
3x - 10y = 10 ----(equation 4)
Now, we have the system of equations:
3x - 10y = 10 ----(equation 4)
-3x + 5y = -20 ----(equation 2)
Next, let's solve this new system using the method of elimination:
Multiply equation 4 by 2:
2(3x - 10y) = 2(10)
6x - 20y = 20 ----(equation 5)
Add equation 2 and equation 5:
(-3x + 5y) + (6x - 20y) = (-20) + 20
3x - 10y = 0
Now, we have:
3x - 10y = 0 ----(equation 6)
3x - 10y = 10 ----(equation 5)
If we examine equation 6, we can see that it is equivalent to the equation -3x + 10y = 0 from equation 1.
Since equation 5 is the same as equation 6, we can say that the two equations are equivalent. This means that the system of equations has infinitely many solutions and that the two equations represent the same line on the coordinate plane.
Therefore, the graph that represents the solution to the given system is the straight line represented by equation -3x + 5y = -20.
Graph of -3x + 5y = -20:
*[illustration graph_of_equation.PNG]