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
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.
To solve the system using elimination, we need to eliminate one of the variables by adding or subtracting the equations.
First, let's multiply the second equation by 2 to make the coefficients of y the same:
2(3x - 3y) = 2(18)
6x - 6y = 36
Now we have the following system of equations:
5x + 4y = 12
6x - 6y = 36
To eliminate y, we can multiply the first equation by 6 and the second equation by 5:
6(5x + 4y) = 6(12)
30x + 24y = 72
5(6x - 6y) = 5(36)
30x - 30y = 180
Now we have the following system of equations:
30x + 24y = 72
30x - 30y = 180
Next, subtract the second equation from the first equation to eliminate x:
(30x + 24y) - (30x - 30y) = 72 - 180
30x + 24y - 30x + 30y = 72 - 180
54y = -108
Divide both sides of the equation by 54:
54y/54 = -108/54
y = -2
Now substitute the value of y back into either of the original equations, let's use the first equation:
5x + 4(-2) = 12
5x -8 = 12
5x = 12 + 8
5x = 20
Divide both sides of the equation by 5:
x = 20/5
x = 4
Therefore, the solution to the system of equations is x = 4 and y = -2.
To solve the system of equations using elimination, we want to eliminate one of the variables.
Let's start by multiplying the second equation by 4 to make the coefficients of y in both equations the same:
4(3x - 3y) = 4(18)
12x - 12y = 72
Now we have two equations:
5x + 4y = 12
12x - 12y = 72
Next, we can add the two equations together to eliminate y:
(5x + 4y) + (12x - 12y) = (12) + (72)
17x - 8y = 84
Now we have a new equation:
17x - 8y = 84
To eliminate y, we can multiply the first equation by 2:
2(5x + 4y) = 2(12)
10x + 8y = 24
Now we have two equations:
10x + 8y = 24
17x - 8y = 84
Adding these two equations together will eliminate y:
(10x + 8y) + (17x - 8y) = (24) + (84)
27x = 108
Dividing both sides of the equation by 27 gives us:
x = 4
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the first equation:
5x + 4y = 12
5(4) + 4y = 12
20 + 4y = 12
4y = 12 - 20
4y = -8
y = -2
So the solution to the system of equations is x = 4 and y = -2.