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 manipulating the equations.
To eliminate x, we need to multiply the second equation by 5 and the first equation by 3:
3(5x + 4y) = 3(123)
5(3x - 3y) = 5(18)
Simplifying:
15x + 12y = 369
15x - 15y = 90
Now, we can subtract the second equation from the first equation:
(15x + 12y) - (15x - 15y) = 369 - 90
15x + 12y - 15x + 15y = 279
Simplifying further:
27y = 279
Dividing both sides of the equation by 27:
y = 279/27
y = 9
Substituting the value of y into one of the original equations, let's use the first equation:
5x + 4(9) = 123
5x + 36 = 123
5x = 123 - 36
5x = 87
x = 87/5
Therefore, the solution to the system of equations is x = 87/5 and y = 9.
To solve the given system of equations using the elimination method, we need to eliminate one variable by adding or subtracting the equations. Here's how we can do that:
1. Multiply both sides of the first equation by 3:
3(5x + 4y) = 3(12)
This simplifies to:
15x + 12y = 36
2. Multiply both sides of the second equation by 5:
5(3x - 3y) = 5(18)
This simplifies to:
15x - 15y = 90
Now we have two equations:
15x + 12y = 36 (Equation A)
15x - 15y = 90 (Equation B)
3. To eliminate the x-term, subtract Equation B from Equation A:
(15x + 12y) - (15x - 15y) = 36 - 90
This simplifies to:
27y = -54
4. Divide both sides by 27 to solve for y:
27y/27 = -54/27
This simplifies to:
y = -2
5. Substitute the value of y = -2 into one of the original equations. Let's use Equation A:
15x + 12(-2) = 36
Simplify this equation:
15x - 24 = 36
6. Add 24 to both sides to isolate the term with x:
15x = 60
7. Divide both sides by 15 to solve for x:
15x/15 = 60/15
This simplifies to:
x = 4
So the solution to the system of equations is x = 4 and y = -2.
To solve the system using elimination, we need to eliminate one of the variables by adding or subtracting the equations.
Let's eliminate the variable y.
To do this, we'll multiply the second equation by 4 to make the coefficients of y-s in both equations equal.
3x - 3y = 18 (Equation 1)
4(3x - 3y) = 4(18)
12x - 12y = 72 (Equation 2)
Now, we can add the two equations together to eliminate y.
5x + 4y = 12
+ 12x - 12y = 72
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17x - 8y = 90 (Equation 3)
Now, let's solve Equation 3 for either x or y.
17x - 8y = 90
Let's solve for x.
17x = 90 + 8y
17x = 8y + 90
x = (8y + 90)/17
Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y.
Let's substitute x in Equation 1.
3( (8y + 90)/17) - 3y = 18
(24y + 270) / 17 - 3y = 18
Now, we can solve for y.
(24y + 270) / 17 - 3y = 18
Multiply all the terms by 17 to eliminate the fraction:
24y + 270 - 51y = 306
Combine like terms:
-27y + 270 = 306
Subtract 270 from both sides:
-27y = 36
Divide both sides by -27:
y = -36/27
Simplify:
y = -4/3
Now that we know the value of y, we can substitute it back into Equation 1 to find the value of x.
3x - 3(-4/3) = 18
3x + 4 = 18
3x = 18 - 4
3x = 14
x = 14/3
Therefore, the solution to the system of equations is:
x = 14/3
y = -4/3