Question

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Solve the system using elimination.

5x + 4y = 123x – 3y = 18

Answer 1

To solve the system using elimination, we will manipulate the equations in order to eliminate one of the variables.

First, we will multiply the second equation by 5 to make the coefficients of x in both equations the same:

5(3x) - 5(3y) = 5(18)
15x - 15y = 90

Now, we will subtract the first equation from this new equation:

(15x - 15y) - (5x + 4y) = 90 - 123
15x - 15y - 5x - 4y = -33
(15x - 5x) + (-15y - 4y) = -33
10x - 19y = -33

Now we have a new equation: 10x - 19y = -33.

Now, we will multiply the first equation by 3 to make the coefficients of x in both equations the same:

3(5x) + 3(4y) = 3(123)
15x + 12y = 369

Next, we will subtract the second equation from this new equation:

(15x + 12y) - (3x - 3y) = 369 - 18
15x + 12y - 3x + 3y = 351
(15x - 3x) + (12y + 3y) = 351
12x + 15y = 351

Now we have a new equation: 12x + 15y = 351.

We now have a system of two linear equations:

10x - 19y = -33
12x + 15y = 351

We will now solve this system using elimination.

Multiplying the first equation by 12 and the second equation by 10, we can create equations with the same coefficients for x:

12(10x - 19y) = 12(-33)
10(12x + 15y) = 10(351)

Expanding and simplifying these equations, we have:

120x - 228y = -396
120x + 150y = 3510

Now, subtracting the first equation from the second equation, we can eliminate x:

(120x + 150y) - (120x - 228y) = 3510 - (-396)
120x + 150y - 120x + 228y = 3510 + 396
378y = 3906
y = 3906/378
y ≈ 10.3

Substituting this value of y back into the first equation:

10x - 19(10.3) = -33
10x - 197.7 = -33
10x = -33 + 197.7
10x ≈ 164.7
x ≈ 164.7/10
x ≈ 16.47

Therefore, the solution to the system is x ≈ 16.47, y ≈ 10.3.