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Question

Solve the system using elimination.

5x + 4y = 123x – 3y = 18

Answer 1

To solve the system using elimination, we need to eliminate one of the variables by adding or subtracting the two equations.

In this case, we can eliminate the variable "y" by multiplying the second equation by 4 to make the coefficients of "y" the same:

5x + 4y = 12 (equation 1)
12x - 12y = 72 (equation 2)

Now we can subtract equation 1 from equation 2 to eliminate "y":

(12x - 12y) - (5x + 4y) = 72 - 12
12x - 12y - 5x - 4y = 60
7x - 16y = 60

Now we have a new equation with only "x" and "y".

Next, we can solve this equation simultaneously with equation 1:

5x + 4y = 12 (equation 1)
7x - 16y = 60 (equation 3)

To eliminate "y", we need to multiply equation 1 by 4 and equation 3 by -1:

20x + 16y = 48 (equation 4)
-7x + 16y = -60 (equation 5)

Now we can add equations 4 and 5 to eliminate "y":

(20x + 16y) + (-7x + 16y) = 48 - 60
20x - 7x + 16y + 16y = -12
13x + 32y = -12

We now have a new equation with only "x" and "y".

Now we can solve this equation simultaneously with equation 2:

13x + 32y = -12 (equation 6)
12x - 12y = 72 (equation 2)

To eliminate "y", we need to multiply equation 6 by 3 and equation 2 by 32:

39x + 96y = -36 (equation 7)
384x - 384y = 2304 (equation 8)

Now we can add equations 7 and 8 to eliminate "y":

(39x + 96y) + (384x - 384y) = -36 + 2304
39x + 384x + 96y - 384y = 2268
423x - 288y = 2268

We now have a new equation with only "x" and "y".

Next, we can solve this equation simultaneously with equation 3:

423x - 288y = 2268 (equation 9)
7x - 16y = 60 (equation 3)

To eliminate "y", we need to multiply equation 9 by 16 and equation 3 by 288:

6768x - 4608y = 36288 (equation 10)
2016x - 4608y = 17280 (equation 11)

Now we can subtract equation 11 from equation 10 to eliminate "y":

(6768x - 4608y) - (2016x - 4608y) = 36288 - 17280
6768x - 2016x - 4608y + 4608y = 19008
4752x = 19008
x = 19008/4752
x = 4

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use equation 1:

5x + 4y = 12
5(4) + 4y = 12
20 + 4y = 12
4y = 12 - 20
4y = -8
y = -8/4
y = -2

Therefore, the solution to the system of equations is x = 4 and y = -2.