Solve the systems of equations by elimination
2x+4y = -8
3x-5y =21
To solve the system of equations by elimination, we can eliminate one variable by multiplying one or both equations by appropriate factors so that the coefficients of one of the variables in both equations become opposites. Let's eliminate the variable "x" by multiplying the first equation by 3 and the second equation by 2:
(3)(2x+4y) = (3)(-8)
(2)(3x-5y) = (2)(21)
Simplifying these equations gives:
6x + 12y = -24
6x - 10y = 42
Now, subtracting the second equation from the first equation, we have:
(6x + 12y) - (6x - 10y) = -24 - 42
6x + 12y - 6x + 10y = -66
22y = -66
y = -66/22
y = -3
Substituting this value of y back into the first equation, we can solve for x:
2x + 4(-3) = -8
2x - 12 = -8
2x = -8 + 12
2x = 4
x = 4/2
x = 2
Therefore, the solution to the system of equations is x = 2 and y = -3.
To solve the system of equations by elimination, we will eliminate one variable by multiplying one or both of the equations by appropriate constants so that the coefficients of one of the variables (x or y) will cancel out when the equations are added or subtracted.
Let's start by multiplying the first equation by 3 and the second equation by 2 to eliminate the x variable:
Equation 1: 2x + 4y = -8 (multiply by 3)
New Equation 1: 6x + 12y = -24
Equation 2: 3x - 5y = 21 (multiply by 2)
New Equation 2: 6x - 10y = 42
Now, subtract the New Equation 2 from New Equation 1:
(6x + 12y) - (6x - 10y) = -24 - 42
Simplifying this equation, we get:
6x + 12y - 6x + 10y = -66
22y = -66
Divide both sides of the equation by 22:
y = -3
Now that we have the value of y, substitute it back into one of the original equations to solve for x. Let's use the first equation:
2x + 4(-3) = -8
2x - 12 = -8
Add 12 to both sides:
2x = 4
Divide both sides by 2:
x = 2
So, the solution to the system of equations is x = 2 and y = -3.
To solve the system of equations by elimination, we need to eliminate one of the variables by adding or subtracting the two equations.
Let's start by multiplying the second equation by 2 so that the coefficients of x will be equal and opposite in both equations.
2(3x-5y) = 2(21)
6x - 10y = 42
Now, we can subtract the first equation from the second equation to eliminate x.
(6x - 10y) - (2x + 4y) = 42 - (-8)
6x - 10y - 2x - 4y = 42 + 8
(6x - 2x) + (-10y - 4y) = 50
4x - 14y = 50
Now we have a new equation. Let's write it along with the first equation:
2x + 4y = -8 (equation 1)
4x - 14y = 50 (equation 2)
We can eliminate y by multiplying the first equation by -7 and the second equation by 2 to make the coefficients of y equal and opposite.
-7(2x + 4y) = -7(-8)
-14x - 28y = 56
2(4x - 14y) = 2(50)
8x - 28y = 100
Now, we can add the new equations to eliminate y.
(-14x - 28y) + (8x - 28y) = 56 + 100
-14x + 8x - 28y - 28y = 156
-6x - 56y = 156
So, the system of equations becomes:
-6x - 56y = 156 (new equation 1)
4x - 14y = 50 (equation 2)
We eliminated y successfully. Now, we can solve these equations by elimination or substitution to find the values of x and y.