To solve the following system of equations by elimination, which step would best to perform first?
1/2a - 1/4b = 1
1/3a + b= 3
A. Add the two equations.
B. Subtract the two equations.
C. Multiply the first equation by 4.
D. Divide the first equation by 3.
I'd do C
To solve the system of equations by elimination, the goal is to manipulate the equations in such a way that one variable cancels out when the equations are added or subtracted.
In this case, the equations involve fractions. To eliminate fractions, we need to get rid of the denominators by finding a common denominator. The least common denominator (LCD) is often used as it simplifies the process.
The LCD of the given fractions in the first equation is 4, as it is the least common multiple of the denominators 2 and 4. The LCD of the fractions in the second equation is 3, as it is the least common multiple of 3 and 1.
To simplify solving the system, we will start by finding the LCD and then using it to manipulate the equations:
1. Multiply the first equation by 4:
(4) * (1/2a - 1/4b) = (4) * 1
2a - b = 4
2. Leave the second equation as it is:
1/3a + b = 3
Now that we have transformed the system of equations, the step that would be best to perform first is option B, subtracting the two equations.
B. Subtract the two equations:
(2a - b) - (1/3a + b) = 4 - 3
2a - b - 1/3a - b = 1
(2a - 1/3a) - (b + b) = 1
By subtracting the two equations, we now have a new equation with only one variable, which allows us to continue solving for that variable using subsequent steps.