Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
-3x + 4y = -19.5
-3x + y = -10.5
-3x + 4y = -19.5
-3x + y = -10.5
-3x + y = -10.5
+3x +3x
y = 3x - 10.5
-3x +4y = -19.5
-3x +4(3x -10.5) = -19.5
-3x + 4(3x) + 4(-10.5) = -16.5
-3x +12x - 42 = -16.5
9x - 42 = -16.5
+42 = + 42
9x = 42 - 19.5
9x = 22.5
9x / 9x= 22.5 / 9
x = 2.5
Plug x into the original equation to solve for y
y = 3x - 10.5
y = 3(2.5) - 10.5
y = 7.5 - 10.5
y = -3
Solution: (2.5, -3)
Check
-3x + 4y = -19.5 -3x + y = -10.5
-3(2.5) + 4(-3) = - 19.5 -3(2.5) + -3 = -10.5
-7.5 + (-12) = -19.5 -7.5 + (-3) = -10.5
-19.5 = -19.5 -10.5 = -10.5
True True
To solve the given system of equations using the elimination method, we will eliminate one of the variables by adding or subtracting the two equations.
Let's start by eliminating the 'x' variable. To do this, we can multiply both sides of the second equation by 3 to make the coefficients of 'x' in both equations the same.
Original equations:
-3x + 4y = -19.5 ----(Equation 1)
-3x + y = -10.5 ----(Equation 2)
Multiply Equation 2 by 3:
-3x + (3)(y) = (3)(-10.5)
-3x + 3y = -31.5 ----(Equation 3)
Now, we will subtract Equation 1 from Equation 3 to eliminate the 'x' variable:
(Equation 3) - (Equation 1):
(-3x + 3y) - (-3x + 4y) = (-31.5) - (-19.5)
-3x + 3y +3x - 4y = -31.5 + 19.5
-y = -12
Simplifying further, we obtain:
-y = -12
To isolate 'y', we can multiply both sides of the equation by -1 to change the sign:
-1(-y) = -1(-12)
y = 12
Now that we have found the value of 'y', we can substitute it back into one of the original equations, let's use Equation 2:
-3x + y = -10.5
Substituting y = 12:
-3x + 12 = -10.5
Subtract 12 from both sides:
-3x = -10.5 - 12
-3x = -22.5
Dividing both sides by -3:
x = -22.5 / -3
x = 7.5
Therefore, the solution to the system of equations is x = 7.5 and y = 12.