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Solve the system using elimination.
5x + 4y = 123x – 3y = 18
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.
To solve the system using elimination, we need to eliminate one of the variables by adding or subtracting the two equations. In this case, let's eliminate the y variable.
First, we'll multiply 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 can add the two equations together:
(5x + 4y) + (12x - 12y) = 123 + 72
5x + 12x + 4y - 12y = 195
17x - 8y = 195
Now we have a new equation in terms of x and y. Let's continue solving the system.
Next, let's multiply the first equation by 3 to make the coefficients of x in both equations the same:
3(5x + 4y) = 3(123)
15x + 12y = 369
Now we can add the new equation to the equation we obtained earlier:
(15x + 12y) + (17x - 8y) = 369 + 195
15x + 17x + 12y - 8y = 564
32x + 4y = 564
Now we have another new equation in terms of x and y.
To eliminate one variable again, let's multiply the third equation by 2:
2(17x - 8y) = 2(195)
34x - 16y = 390
Now we can subtract this equation from the equation we obtained earlier:
(32x + 4y) - (34x - 16y) = 564 - 390
32x - 34x + 4y + 16y = 174
-2x + 20y = 174
Now we have one final equation in terms of x and y.
To solve this equation, we can add it to the equation 17x - 8y = 195, since we can eliminate the y variable now:
(-2x + 20y) + (17x - 8y) = 174 + 195
-2x + 17x + 20y - 8y = 369
15x + 12y = 369
We now have our final equation in terms of x and y.
To solve for x, we can isolate x in the last equation:
15x = 369 - 12y
x = (369 - 12y) / 15
To solve for y, we can substitute the value of x we just found into any of the previous equations. Let's use the third equation:
17x - 8y = 195
17((369 - 12y) / 15) - 8y = 195
Simplifying this equation will give us the value of y.
Therefore, the solution to the system of equations is x = (369 - 12y) / 15 and y = some value obtained by solving the simplified equation.