Solve the system using elimination.
10y − 2x = 50
6x − 3y = 12
A. (5, 6)
B. (2.1, 0.2)
C. (2.1, 4.6)
D. (6, 5)
multipy the second equation by 10/6
10x-5y=20
now subtract the two equation.
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To solve the system of equations using elimination, we need to eliminate one variable by adding or subtracting the equations.
Let's start by multiplying the two equations to make the coefficients of x or y in one equation equal to the coefficients in the other equation.
First, let's multiply the first equation by -3:
-3(10y - 2x) = -3(50)
-30y + 6x = -150
Next, let's multiply the second equation by 2:
2(6x - 3y) = 2(12)
12x - 6y = 24
Now we have:
-30y + 6x = -150
12x - 6y = 24
Next, let's add the two equations together to eliminate y:
(-30y + 6x) + (12x - 6y) = -150 + 24
-30y + 6x + 12x - 6y = -126
18x - 36y = -126
Now we can divide both sides of the equation by 18 to solve for x:
(18x - 36y) / 18 = -126 / 18
x - 2y = -7
Now we have two equations:
x - 2y = -7 (Equation 1)
12x - 6y = 24 (Equation 2)
To simplify Equation 1, let's multiply it by 6:
6(x - 2y) = 6(-7)
6x - 12y = -42
Now we can subtract Equation 2 from Equation 3 to eliminate x:
(6x - 12y) - (12x - 6y) = -42 - 24
6x - 12y - 12x + 6y = -66
-6x - 6y = -66
Now we can divide both sides of the equation by -6 to solve for y:
(-6x - 6y) / -6 = -66 / -6
x + y = 11
Now we have two equations:
x + y = 11 (Equation 4)
x - 2y = -7 (Equation 1)
To solve for x and y, we can add equations 1 and 4 together:
(x + y) + (x - 2y) = 11 + (-7)
2x - y = 4
Now we can solve this equation to find the value of x:
2x - y = 4
2x = y + 4
x = (y + 4) / 2
Now we can substitute this value of x into Equation 4 to solve for y:
x + y = 11
((y + 4) / 2) + y = 11
(y + 4) + 2y = 22
3y + 4 = 22
3y = 18
y = 6
Now we can substitute the value of y back into Equation 4 to solve for x:
x + y = 11
x + 6 = 11
x = 11 - 6
x = 5
Therefore, the solution to the system of equations is (5, 6), which corresponds to option A.
To solve the system of equations using elimination, we need to eliminate one of the variables by multiplying one or both equations by a suitable factor. Let's follow these steps:
Step 1: Multiply the first equation by 6, and the second equation by 10 to make the coefficients of "x" the same for both equations:
6(10y - 2x) = 6(50) --> 60y - 12x = 300
10(6x - 3y) = 10(12) --> 60x - 30y = 120
Step 2: Now we have two equations that have the same coefficient for "x." We can add these two equations together to eliminate the "x" variable:
(60y - 12x) + (60x - 30y) = 300 + 120
60y - 12x + 60x - 30y = 420
-18x + 30y = 420
Step 3: Simplify the equation:
-18x + 30y = 420
Step 4: Divide the equation by 6 to get simpler coefficients:
(-18/6)x + (30/6)y = 420/6
-3x + 5y = 70 --> Equation 1
Step 5: Now let's use the second equation from the original system:
6x - 3y = 12 --> Equation 2
Step 6: Multiply Equation 2 by 2 to make the coefficients of "y" the same and add it to Equation 1:
2(6x - 3y) + (-3x + 5y) = 2(12) + 70
12x - 6y - 3x + 5y = 24 + 70
9x - y = 94
Step 7: Now we have eliminated the "y" variable. Simplify this equation:
9x - y = 94 --> Equation 3
Step 8: Solve Equations 2 and 3 as a system of equations:
Equation 2: 6x - 3y = 12
Equation 3: 9x - y = 94
To solve this system further, we need to use either substitution or elimination again. However, it seems that there may be a mistake or inconsistency in the original system of equations. Therefore, we cannot determine the solution to this system using elimination.