Enter your answer and show all the steps that you use to solve this problem in the space provided. Solve the system using elimination. 5x + 4y = 123x – 3y = 18

Answer 1

To solve this system of equations using elimination, we'll multiply one of the equations by a constant so that the coefficients of either x or y are the same in both equations.

Let's start by multiplying the second equation by 4 to eliminate y:

(4)(3x - 3y) = (4)(18)
12x - 12y = 72

Now, we have two equations:

5x + 4y = 123
12x - 12y = 72

Next, we'll add the two equations together to eliminate y:

(5x + 4y) + (12x - 12y) = 123 + 72
5x + 12x + 4y - 12y = 195
17x - 8y = 195

Now we have a new equation:

17x - 8y = 195

To solve for x, we can isolate it by multiplying the second equation by -17 and adding it to the new equation:

(-17)(5x + 4y) + (17)(17x - 8y) = (-17)(123) + (17)(195)
-85x - 68y + 289x - 136y = -2091 + 3315
204x - 204y = 1224

Simplifying this equation, we get:

204x - 204y = 1224

Now, we have two equations:

17x - 8y = 195
204x - 204y = 1224

To eliminate y, we can multiply the first equation by 25 and the second equation by 2:

(25)(17x - 8y) = (25)(195)
(2)(204x - 204y) = (2)(1224)

Simplifying these equations, we have:

425x - 200y = 4875
408x - 408y = 2448

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

(425x - 200y) - (408x - 408y) = 4875 - 2448
425x - 200y - 408x + 408y = 2427
17x + 208y = 2427

Finally, we have a new equation:

17x + 208y = 2427

To solve for x, we can isolate it by multiplying the first equation by -208 and adding it to the new equation:

(-208)(17x + 208y) + (208)(17x + 208y) = (-208)(2427) + (208)(2427)
-3536x - 43328y + 3536x + 43328y = -504216 + 504216
0 = 0

Since 0 = 0, this means that the equations are dependent, and there are infinitely many solutions. In other words, any pair of numbers (x, y) that satisfies one equation will also satisfy the other equation.

Hence, the system has infinitely many solutions.