solve using the elimination method. show your work. if the system has no solution or an infinte number of solutions, state this.
14x+18y=-54
-4x-14y=42
help me please
Eq1: 14x + 18y = -54.
Eq2: -4x - 14y = 42.
Divide both sides of each Eq by 2:
Eq1: 7x + 9y = --27.
Eq2: -2x - 7y = 21.
Multiply Eq1 by 2 and Eq2 by 7.
14x + 18y = -54
-14x - 49y = 147
Sum:0-31y = 93
Y = -3.
In Eq1, substitute -3 for Y:
14x + 18(-3) = -54
14x - -54 = -54
14x = -54 + 54 = 0
X = 0.
Solution set: (x,y) = (0,-3).
To solve the given system of equations using the elimination method, we need to eliminate one of the variables by manipulating the equations and then solve for the remaining variable.
Let's start by multiplying the second equation by 7 to make the coefficients of y in both equations equal:
-4x - 14y = 42 --> -28x - 98y = 294
Now, we can add the two equations together:
14x + 18y = -54
-28x - 98y = 294
------------------
-14x - 80y = 240
Now, we have a new equation:
-14x - 80y = 240
Next, let's divide this equation by -2 to simplify it:
-14x / -2 - 80y / -2 = 240 / -2
7x + 40y = -120
Now, we have a new equation:
7x + 40y = -120
Let's rewrite the original first equation:
14x + 18y = -54
Now, let's multiply it by -2 to make the coefficients of x in both equations equal:
-2(14x + 18y) = -2(-54)
-28x - 36y = 108
Now, we can add this equation to (-14x - 80y = 240):
(-28x - 36y) + (-14x - 80y) = 108 + 240
-42x - 116y = 348
Now, we have a new equation:
-42x - 116y = 348
We have simplified the system of equations, but we can see that the coefficients of x and y are still not the same in the two equations.
So, this system of equations has no solution because the lines represented by the equations are parallel and do not intersect.