Solve the system by the elimination method.
4x−3y=
2
8x+12y=
10
.
.
.
Question content area right
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is enter your response here.
(Type an ordered pair. Type integers or simplified fractions.)
B.
There are infinitely many solutions.
The solution set is x,enter your response here, where x is any real number. (Type integers or simplified fractions.)
C.
The solution is the empty set.
To solve the given system of equations by the elimination method, we'll eliminate one variable by multiplying the equations by appropriate numbers so that the coefficients of one variable in both equations are equal.
Given equations:
1) 4x - 3y = 2
2) 8x + 12y = 10
To eliminate the y variable, we can multiply equation 1) by 4 and equation 2) by 3.
3) 16x - 12y = 8
4) 24x + 36y = 30
Now, we can add equations 3) and 4) to eliminate the y variable:
(16x - 12y) + (24x + 36y) = 8 + 30
40x + 24y = 38
The new equation obtained is:
5) 40x + 24y = 38
Now, let's divide the equation by 2 to simplify it:
(40x + 24y)/2 = 38/2
20x + 12y = 19
Now, we have two equations:
5) 20x + 12y = 19
2) 8x + 12y = 10
We can subtract equation 2) from equation 5) to eliminate the y variable:
(20x + 12y) - (8x + 12y) = 19 - 10
20x - 8x + 12y - 12y = 9
12x = 9
Dividing both sides of the equation by 12:
x = 9/12
x = 3/4
Substituting this value of x into equation 2):
8(3/4) + 12y = 10
6 + 12y = 10
12y = 10 - 6
12y = 4
y = 4/12
y = 1/3
Therefore, the solution to the given system of equations is the ordered pair:
(x, y) = (3/4, 1/3)
So, the correct choice is A. The solution set is (3/4, 1/3).
To solve the system by the elimination method, we can multiply the first equation by 6 and the second equation by 3 to eliminate the y variable:
6(4x - 3y) = 6(2) (Multiply the first equation by 6)
3(8x + 12y) = 3(10) (Multiply the second equation by 3)
This simplifies the system to:
24x - 18y = 12
24x + 36y = 30
Now, subtract the second equation from the first equation to eliminate the x variable:
(24x - 18y) - (24x + 36y) = 12 - 30
This simplifies to:
-54y = -18
Divide both sides of the equation by -54:
y = 18/54
Simplifying the fraction by dividing both the numerator and denominator by 18:
y = 1/3
Now substitute this value of y into either of the original equations, such as the first equation:
4x - 3(1/3) = 2
Simplifying the equation:
4x - 1 = 2
Add 1 to both sides of the equation:
4x = 3
Divide both sides of the equation by 4:
x = 3/4
Therefore, the solution to the system is (x, y) = (3/4, 1/3).
So, the correct choice is A. The solution set is (3/4, 1/3).
To solve the given system of equations using the elimination method, we first need to eliminate one variable by adding or subtracting the equations.
Given system of equations:
1) 4x - 3y = 2
2) 8x + 12y = 10
Let's manipulate the equations so that the coefficients of either x or y are the same in both equations. In this case, we'll eliminate the y variable.
Multiply equation 1 by 4 and equation 2 by 3 to make the coefficients of y the same:
1) 16x - 12y = 8
2) 24x + 36y = 30
Now, we can add equation 1 and equation 2:
(16x - 12y) + (24x + 36y) = 8 + 30
40x + 24y = 38
Simplify the equation:
4(10x + 6y) = 38
40x + 24y = 38
Now we have a new equation:
3) 40x + 24y = 38
Comparing equation 3 to equation 2, we can see that they represent the same line (each side can be divided by 2 to obtain the same equation). This means that the original system of equations is dependent, and there are infinitely many solutions.
Therefore, the correct choice is:
B. There are infinitely many solutions. The solution set is x, enter your response here, where x is any real number. (Type integers or simplified fractions.)