solve the system of equations
5x-2y=18
3x+3y=15
To solve the system of equations, we can use either the substitution method or the elimination method. Let's use the elimination method.
First, let's multiply the second equation by 2 to eliminate the y term:
2(3x + 3y) = 2(15)
6x + 6y = 30
Next, let's subtract this new equation from the first equation:
(5x - 2y) - (6x + 6y) = 18 - 30
5x - 2y - 6x - 6y = -12
Simplifying this equation, we get:
-x - 8y = -12
Now, let's multiply the second equation by -2 to eliminate the x term:
-2(3x + 3y) = -2(15)
-6x - 6y = -30
Adding this new equation to the equation we found earlier, we get:
(-x - 8y) + (-6x - 6y) = -12 - 30
-x - 8y - 6x - 6y = -42
- 7x - 14y = -42
Now, let's multiply this equation by -1 to simplify:
7x + 14y = 42
Now we have a new system of equations:
-7x - 14y = -42
7x + 14y = 42
Adding these two equations together, we get:
-7x - 14y + 7x + 14y = -42 + 42
0 = 0
This equation is always true, which means that the two original equations are dependent. This means that the solution set is infinitely many points that satisfy both equations. In other words, every (x, y) that satisfies one equation will also satisfy the other equation.
Therefore, the system of equations has infinitely many solutions.