Im studying for finals!
A system of equations is shown below:
8x + 5y = 9 3x + 2y = 4
Part A: Create an equivalent system of equations by replacing one equation with the sum of that equation and a multiple of the other. Show the steps to do this.
Part B: Show that the equivalent system has the same solution as the original system of equations.
A: let's add them as suggested:
11x + 7y = 13
multiply the 2nd by 2
6x + 4y = 8
Note: the system : 11x+7y = 13 and 6x + 4y = 8 is not unique, I could have done anything I wanted here.
B.
ok, let's solve the original system:
#1 times 2: 16x + 10y = 18
#2 times 5: 15x + 10y = 20
subtract them:
x = -2
back in the original #1
8(-2) + 5y = 9
5y = 25
y = 5
so x = -2, y = 5
test these in our new system of A
11x + 7y = 13
11(-2) + 7(5) = -22 + 35 = 13, Yeah!
6x + 4y = 8
6(-2) + 4(5) = -12 + 20 = 8, yeahhhhh!!!!
Okay so for part A I should put the answer is 6x + 4y = 8 ?
No,
you want a "system", which means you need two equations.
My new system was:
11x + 7y = 13 and 6x + 4y = 8
The original 8x + 5y = 9 was replaced by the sum of the two original equations,
and the original 3x + 2y = 4 was replaced by a multiple of that (I multiplied it by 2)
I just followed their instructions.
Part A: To create an equivalent system of equations by replacing one equation with the sum of that equation and a multiple of the other, we need to choose one of the equations and multiply it by a constant, and then add it to the other equation.
Let's choose the first equation, 8x + 5y = 9, and multiply it by 2. This gives us:
2(8x + 5y) = 2(9)
Simplifying this, we have:
16x + 10y = 18
Now, we add this new equation to the second equation, 3x + 2y = 4:
(16x + 10y) + (3x + 2y) = 18 + 4
Combining like terms, we get:
19x + 12y = 22
So, the equivalent system of equations is:
19x + 12y = 22
3x + 2y = 4
Part B: To show that the equivalent system has the same solution as the original system of equations, we need to solve both systems and compare the solutions.
Let's solve the original system of equations:
8x + 5y = 9
3x + 2y = 4
We can solve this system using different methods, such as substitution or elimination. Let's use the elimination method in this case.
Multiplying the second equation by 4, we get:
4(3x + 2y) = 4(4)
12x + 8y = 16
Now, we can subtract this new equation from the first equation:
(8x + 5y) - (12x + 8y) = 9 - 16
Simplifying, we get:
-4x - 3y = -7
So, now we have the system:
-4x - 3y = -7
3x + 2y = 4
Solving this system, we find that x = 1 and y = 2.
Now, let's solve the equivalent system of equations:
19x + 12y = 22
3x + 2y = 4
Again, using the elimination method, we can multiply the second equation by 6:
6(3x + 2y) = 6(4)
18x + 12y = 24
Subtracting this new equation from the first equation, we have:
(19x + 12y) - (18x + 12y) = 22 - 24
Simplifying, we get:
x = -2
Substituting this value of x into the second equation, we get:
3(-2) + 2y = 4
Simplifying further, we have:
-6 + 2y = 4
Adding 6 to both sides, we get:
2y = 10
Dividing both sides by 2, we have:
y = 5
So, the solution to the equivalent system is x = -2 and y = 5.
Comparing the solutions, we see that the original system and the equivalent system have different solutions. Therefore, the equivalent system does not have the same solution as the original system of equations.