Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this. 14x + 18y = -54,-4x – 14y = 42
To solve the given system of equations using the elimination method, we need to eliminate one variable by manipulating the two equations. Here's how we can do it:
1. Start with the two equations:
14x + 18y = -54 ---(Equation 1)
-4x - 14y = 42 ---(Equation 2)
2. Multiply Equation 2 by 7 to make the coefficients of y the same but with opposite signs:
-4x - 14y = 42 Multiply by 7
-28x - 98y = 294 ---(Equation 3)
3. Now, add Equation 1 and Equation 3 to eliminate the y variable:
(14x + 18y) + (-28x - 98y) = (-54) + (294)
-14x - 80y = 240 ---(Equation 4)
4. Simplify Equation 4:
-14x - 80y = 240 Divide by -2
7x + 40y = -120 ---(Equation 5)
5. Multiply Equation 1 by 4 to make the coefficients of x the same but with opposite signs:
(14x + 18y) * 4 = (-54) * 4
56x + 72y = -216 ---(Equation 6)
6. Now, add Equation 5 and Equation 6 to eliminate the x variable:
(56x + 72y) + (7x + 40y) = (-216) + (-120)
63x + 112y = -336 ---(Equation 7)
7. Simplify Equation 7:
63x + 112y = -336 Divide by 7
9x + 16y = -48 ---(Equation 8)
8. Notice that Equation 5 and Equation 8 are identical, just multiplied by different constants. This indicates that the two equations are the same line and have an infinite number of solutions.
Therefore, the system of equations has an infinite number of solutions.
14x + 18y = -54
-4x – 14y = 42
multiply top by 2, bottom by 7, and add:
28x + 36y = -108
-28x - 98y = 294
-62y = 186
y = -3
so, x = 0