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Solve the system using elimination.
5x + 4y = 12
3x – 3y = 18
(3 points)
To solve the system using elimination, we need to eliminate one of the variables by multiplying one or both equations by appropriate constants. In this case, we can eliminate the x variable by multiplying the second equation by 5 and the first equation by 3. This will make the coefficients of x in both equations the same and allow us to eliminate x by subtracting the two equations.
First, let's multiply the second equation by 5:
5(3x - 3y) = 5(18)
15x - 15y = 90
Next, let's multiply the first equation by 3:
3(5x + 4y) = 3(12)
15x + 12y = 36
Now we have the system of equations:
15x + 12y = 36
15x - 15y = 90
To eliminate x, we can subtract the second equation from the first equation:
(15x + 12y) - (15x - 15y) = 36 - 90
This simplifies to:
15x + 12y - 15x + 15y = -54
Combining like terms:
27y = -54
To solve for y, divide both sides of the equation by 27:
27y/27 = -54/27
y = -2
Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first equation:
5x + 4(-2) = 12
Simplifying:
5x - 8 = 12
To solve for x, add 8 to both sides:
5x = 12 + 8
5x = 20
Divide both sides of the equation by 5:
x = 20/5
x = 4
So the solution to the system of equations is x = 4 and y = -2.
To solve the system of equations using elimination, we will eliminate one of the variables by adding the two equations together. Let's start by multiplying the second equation by 4 to make the coefficient of y opposite:
Original equations:
5x + 4y = 12 - Equation (1)
3x - 3y = 18 - Equation (2)
Multiply equation (2) by 4:
4(3x - 3y) = 4(18)
12x - 12y = 72 - Equation (3)
Next, we will add equation (1) and equation (3):
(5x + 4y) + (12x - 12y) = 12 + 72
Combine like terms:
5x + 12x + 4y - 12y = 84
Simplify:
17x - 8y = 84
Now we have a new equation, let's call it equation (4), which is obtained by adding equations (1) and (3) together.
So the system of equations becomes:
17x - 8y = 84 - Equation (4)
3x - 3y = 18 - Equation (2)
To eliminate the y variable, we need to make the y coefficients in both equations equal. Multiply equation (4) by 3 and multiply equation (2) by 8:
3(17x - 8y) = 3(84)
8(3x - 3y) = 8(18)
Simplify:
51x - 24y = 252 - Equation (5)
24x - 24y = 144 - Equation (6)
Now, subtract equation (6) from equation (5) to eliminate the y variable:
(51x - 24y) - (24x - 24y) = 252 - 144
Combine like terms:
51x - 24x - 24y + 24y = 108
Simplify:
27x = 108
Next, we isolate x by dividing both sides of the equation by 27:
27x/27 = 108/27
Simplify:
x = 4
Now that we have the value of x, we can substitute it back into equation (2) to solve for y:
3x - 3y = 18
Substitute x = 4:
3(4) - 3y = 18
Simplify:
12 - 3y = 18
Subtract 12 from both sides:
-3y = 6
Now, divide both sides by -3 to solve for y:
y = -2
Therefore, the solution to the system of equations is x = 4 and y = -2.