Solve using two different methods. Explain which method you found to be most efficient.
A.
3x - 9y = 3
6x - 3y = -24
B.
7x - 3y = 20
5x + 3y = 16
C.
y = 1/2x - 6
2x + 6y = 19
I am just stuck and don't know how to do these last few problems on my assignment. If someone would please help it would be very much appreciated. Thank You.
A. Elimination method
-6x +18y = -6
6x -3y = -24
15y = -30
y = -2
x = -5
(-5,-2)
B. Elimination method
7x -3y = 20
5x +3y = 16
12x = 36
x = 3
y = 1/3
(3, 1/3)
C . Substitution method
y =x/2 -6
2x + 6y = 19
2x + 6 (x/2 -6) = 19
2x + 3x -36 = 19
5x -36 = 19
5x = 55
x = 11
y = -1/2
(11, -1/2)
Thank you so much. But I need 2 ways to solve each one, can you show me another method for each.
I will give you the substitution method for b. Then you do a and c
B. substitution method
7x -3y = 20
7 x = 3 y + 20
x = (3/7)y + 20/7
then
5x +3y = 16
5[ (3/7)y + 20/7 ] + 3 y = 16
(15/7)y + 100/7 + 21/7 y = 112/7
36 y = 12
y = 1/3
go back and get x
Ok. Thank you.
Sure! I'd be happy to help you solve these equations.
A. In this case, we have two equations with two variables:
3x - 9y = 3
6x - 3y = -24
One method to solve this system of linear equations is by the method of substitution.
1. From the first equation, let's isolate x:
3x = 9y + 3
x = (9y + 3) / 3
x = 3y + 1
2. Now, let's substitute this value of x into the second equation:
6(3y + 1) - 3y = -24
18y + 6 - 3y = -24
15y = -30
y = -2
3. Substitute the value of y back into the first equation to solve for x:
3x - 9(-2) = 3
3x + 18 = 3
3x = -15
x = -5
So, the solution to this system of equations is x = -5 and y = -2.
B. For this set of equations:
7x - 3y = 20
5x + 3y = 16
Another method to solve this system of equations is by adding or subtracting the equations together.
1. Let's add the two equations:
(7x - 3y) + (5x + 3y) = 20 + 16
12x = 36
x = 3
2. Now, substitute the value of x back into either of the original equations to solve for y. Let's use the first equation:
7(3) - 3y = 20
21 - 3y = 20
-3y = -1
y = 1/3
So, the solution to this system of equations is x = 3 and y = 1/3.
C. For this set of equations:
y = 1/2x - 6
2x + 6y = 19
We can use the method of substitution to solve them.
1. Substitute the value of y from the first equation into the second equation:
2x + 6(1/2x - 6) = 19
2x + 3x - 36 = 19
5x = 55
x = 11
2. Now, substitute the value of x back into the first equation to solve for y:
y = 1/2(11) - 6
y = 11/2 - 6
y = 11/2 - 12/2
y = -1/2
So, the solution to this system of equations is x = 11 and y = -1/2.
In terms of efficiency, it depends on the specific equations and the skills of the person solving them. Method of substitution and adding/subtracting the equations are both valid and useful. Some equations may have coefficients that lend themselves better to one method over the other. In general, it's helpful to try both methods and choose the one that seems simpler and easier to work with for a specific set of equations.