Note: 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 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.
To solve the system of equations using elimination, we need to eliminate one variable by multiplying both equations by appropriate numbers so that the coefficients of one variable in both equations will be the same.
Let's eliminate the variable "y" by multiplying the first equation by 3 and the second equation by 4. This will make the coefficients of "y" in both equations the same.
Multiply the first equation by 3:
3 * (5x + 4y) = 3 * 123
15x + 12y = 369
Multiply the second equation by 4:
4 * (3x - 3y) = 4 * 18
12x - 12y = 72
Now, we have the system of equations:
15x + 12y = 369
12x - 12y = 72
Add the two equations together to eliminate "y":
(15x + 12y) + (12x - 12y) = 369 + 72
15x + 12y + 12x - 12y = 441
27x = 441
Divide both sides of the equation by 27 to solve for "x":
27x / 27 = 441 / 27
x = 16.333...
Now, substitute the value of x into one of the original equations to solve for "y". Let's use the first equation:
5x + 4y = 123
5(16.333...) + 4y = 123
81.666... + 4y = 123
4y = 41.333...
Divide both sides of the equation by 4 to solve for "y":
4y / 4 = 41.333... / 4
y = 10.333...
Therefore, the solution to the system of equations is x ≈ 16.333 and y ≈ 10.333.
To solve the system of equations using elimination, we need to eliminate one variable by adding or subtracting the equations.
Let's start by eliminating one variable. In this case, let's eliminate the y variable.
To do that, we need to multiply each equation by a suitable number so that the coefficients of y have the same absolute value but different signs.
Looking at the coefficients of y in the two equations:
The coefficient of y in the first equation is 4.
The coefficient of y in the second equation is -3.
To make the absolute values equal, we need to multiply the first equation by 3 and the second equation by 4.
By multiplying the first equation by 3, we get:
(3)(5x + 4y) = (3)(123x - 3y)
15x + 12y = 369x - 9y
And by multiplying the second equation by 4, we get:
(4)(3x - 3y) = (4)(18)
12x - 12y = 72
Now, we can add the two equations together to eliminate the y variable:
(15x + 12y) + (12x - 12y) = (369x - 9y) + 72
15x + 12y + 12x - 12y = 369x - 9y + 72
Simplifying the equation gives:
27x = 369x + 72 - 9y
Next, let's isolate the x variable by subtracting 369x from both sides:
27x - 369x = 369x + 72 - 9y - 369x
-342x = 72 - 9y
And simplify further:
-342x = -9y + 72
To solve for x, divide both sides by -342:
x = (-9y + 72) / -342
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 substitute x = (-9y + 72) / -342 into the first equation:
5x + 4y = 123
Substituting x gives:
5((-9y + 72) / -342) + 4y = 123
Now we can solve for y by simplifying the equation and solving for y:
((-45y + 360) / -342) + 4y = 123
To get rid of the fractions, we can multiply the entire equation by -342:
-342((-45y + 360) / -342) + (-342)(4y) = -342(123)
Simplifying gives:
-45y + 360 - 1368y = -42126
Combine like terms:
-1413y + 360 = -42126
Next, we isolate the y variable by subtracting 360 from both sides:
-1413y = -42126 - 360
Simplifying:
-1413y = -42486
Finally, solve for y by dividing both sides by -1413:
y = -42486 / -1413
Simplifying further:
y = 30
So, the solution to the system of equations is x = (-9y + 72) / -342 and y = 30.