Solve the following system of equations by the elimination method.
1/3x+1/4y=7
1/3x-3/4y=-9
a)The solution is___?(Type an ordered pair. Use integers or fractions for any numbers in the expression)
b)Infinitely many solutions
c)No solution
1/3x + 1/4y = 7
1/3x - 3/4y = -9
Make one of the equations negative to eliminate the x's.
-(1/3x + 1/4y) = -(7)
-1/3x - 1/4y = -7
1/3x - 3/4y = -9
____________________
-y = -16
y = 16
Now solve for x. Use either equation.
1/3x + 1/4(16) = 7
1/3x + 4 = 7
Eliminate the fraction by multiplying the equation by three.
3(1/3x + 4) = 3(7)
x + 12 = 21
x = 21 - 12
x = 9
Solution:
(9, 16)
thank you
To solve the system of equations using the elimination method, we need to eliminate one variable by multiplying one or both equations by appropriate constants so that when the equations are added or subtracted, one variable is eliminated.
Let's start by multiplying both sides of the first equation by 12 to eliminate the fractions:
12 * (1/3x) + 12 * (1/4y) = 12 * 7
4x + 3y = 84
Next, multiply both sides of the second equation by 12 to eliminate the fractions:
12 * (1/3x) - 12 * (3/4y) = 12 * (-9)
4x - 9y = -108
Now, we can subtract the second equation from the first equation:
(4x + 3y) - (4x - 9y) = 84 - (-108)
4x + 3y - 4x + 9y = 84 + 108
12y = 192
Divide both sides of the equation by 12 to solve for y:
(12y)/12 = 192/12
y = 16
Substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
1/3x + 1/4(16) = 7
1/3x + 4 = 7
1/3x = 7 - 4
1/3x = 3
Multiply both sides by 3 to solve for x:
3 * (1/3x) = 3 * 3
x = 9
Therefore, the solution to the system of equations is (x, y) = (9, 16).
a) The solution is (9, 16)
To solve the system of equations using the elimination method, we need to eliminate one variable by adding or subtracting the two equations. Let's start by eliminating the x variable.
Given equations:
1/3x + 1/4y = 7 ----(1)
1/3x - 3/4y = -9 ----(2)
To cancel out the x terms, we can subtract equation (2) from equation (1).
(1) - (2):
(1/3x + 1/4y) - (1/3x - 3/4y) = 7 - (-9)
Simplifying the equation gives us:
(1/3x - 1/3x) + (1/4y + 3/4y) = 7 + 9
The x terms cancel out, leaving us with:
(1/4y + 3/4y) = 16
Combining like terms on the left side:
(4/4y + 12/4y) = 16
(16/4y) = 16
Now, we can solve for y by multiplying both sides by the reciprocal of 16/4 (which is 4/16 or 1/4):
(16/4y) * (1/4) = 16 * (1/4)
(16/16y) = 4
y = 4
We have found the value of y, which is 4. To find the value of x, we substitute this value of y into either of the original equations. Let's use equation (1):
1/3x + 1/4(4) = 7
1/3x + 1 = 7
1/3x = 7 - 1
1/3x = 6
To isolate x, multiply both sides of the equation by 3:
(1/3x) * 3 = 6 * 3
x = 18
Therefore, the solution to the system of equations is the ordered pair (x, y) = (18, 4).
a) The solution is (18, 4).