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Solve the system using elimination.
5x + 4y = 12
3x - 3y = 18
To solve the system using elimination, we need to eliminate one variable by adding or subtracting the two equations.
First, let's multiply the second equation by 4 to make the coefficients of y the same in both equations:
4(3x - 3y) = 4(18)
12x - 12y = 72
Now the system becomes:
5x + 4y = 12
12x - 12y = 72
Next, let's add the two equations together to eliminate y:
(5x + 4y) + (12x - 12y) = 12 + 72
5x + 12x + 4y - 12y = 84
17x - 8y = 84
Now we have a new equation: 17x - 8y = 84.
Let's rearrange this equation to solve for x:
17x = 8y + 84
x = (8y + 84)/17
Now we substitute this value of x into one of the original equations, let's use the first equation:
5( (8y + 84)/17 ) + 4y = 12
Now we solve for y:
(40y + 420)/17 + 4y = 12
(40y + 420) + 68y = 12*17
40y + 420 + 68y = 204
108y + 420 = 204
108y = 204 - 420
108y = -216
y = -216/108
y = -2
Now we substitute this value of y back into the equation we found for x:
x = (8(-2) + 84)/17
x = ( -16 + 84)/17
x = 68/17
x = 4
Therefore, the solution to the system of equations is x = 4, y = -2.
To solve the system using elimination, we want to eliminate one of the variables by manipulating the equations.
First, let's multiply the second equation by 4 so that the coefficients of y in both equations will be equal:
4(3x - 3y) = 4(18)
12x - 12y = 72
Now, our system of equations becomes:
5x + 4y = 12
12x - 12y = 72
Next, we can add the two equations together to eliminate the y variable:
(5x + 4y) + (12x - 12y) = 12 + 72
17x - 8y = 84
Now, we have one equation with only x variables. Next, we can solve for x:
17x - 8y = 84
To solve for x, we will multiply the equation by 8 to eliminate the coefficient of y:
8(17x - 8y) = 8(84)
136x - 64y = 672
Now, our equation becomes:
136x - 64y = 672
To eliminate y, we can multiply the first equation by 64:
64(5x + 4y) = 64(12)
320x + 256y = 768
Now, our equation becomes:
320x + 256y = 768
Next, we can add the two equations together to eliminate the y variable:
(136x - 64y) + (320x + 256y) = 672 + 768
456x = 1440
Now, we can solve for x by dividing both sides of the equation by 456:
456x/456 = 1440/456
x = 3.158
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y.
Let's use the first equation:
5x + 4y = 12
Substituting x = 3.158:
5(3.158) + 4y = 12
15.79 + 4y = 12
4y = 12 - 15.79
4y = -3.79
y = -0.948
Therefore, the solution to the system of equations is x = 3.158 and y = -0.948.
To solve this system of equations using elimination, we want to eliminate one variable by adding or subtracting the equations. Let's start by multiplying the second equation by 4 to make the coefficients of y in both equations the same.
Original system:
Equation 1: 5x + 4y = 12
Equation 2: 3x - 3y = 18
Multiply Equation 2 by 4:
4(3x - 3y) = 4(18)
12x - 12y = 72
Now we have:
Equation 1: 5x + 4y = 12
Equation 2: 12x - 12y = 72
Next, we can add the equations together to eliminate the y term.
Add Equation 1 and Equation 2:
(5x + 4y) + (12x - 12y) = 12 + 72
5x + 12x + 4y - 12y = 84
17x - 8y = 84
Now we have:
Equation 3: 17x - 8y = 84
To further simplify, let's multiply Equation 1 by 2 to make the coefficients of y the same as in Equation 3.
Multiply Equation 1 by 2:
2(5x + 4y) = 2(12)
10x + 8y = 24
Now we have:
Equation 1: 10x + 8y = 24
Equation 3: 17x - 8y = 84
We can add these two equations together to eliminate the y term.
Add Equation 1 and Equation 3:
(10x + 8y) + (17x - 8y) = 24 + 84
10x + 17x + 8y - 8y = 108
27x = 108
Divide both sides by 27 to solve for x:
27x/27 = 108/27
x = 4
Now that we have x, we can substitute it back into one of the original equations to solve for y. Let's use Equation 1.
5x + 4y = 12
5(4) + 4y = 12
20 + 4y = 12
4y = 12 - 20
4y = -8
y = -8/4
y = -2
So the solution to the system of equations is x = 4 and y = -2.