Solve the following system of equations using elimination.
5x+y=24 ; -5x-4y=-56. Solve for X/Y.
Please help me-?
adding the two equations eliminates the x term
y - 4y = 24 - 56 ... solve for y , then substitute back to find x
Sure! I can help you solve the system of equations using elimination.
First, let's multiply the first equation by 4 and the second equation by 1 to get the coefficients of x in both equations to be the same:
4(5x+y) = 4(24)
1(-5x-4y) = 1(-56)
This simplifies to:
20x + 4y = 96
-5x - 4y = -56
Now, let's add these two equations together to eliminate the y term:
(20x + 4y) + (-5x - 4y) = 96 + (-56)
Simplifying further:
20x - 5x + 4y - 4y = 96 - 56
Combining like terms:
15x + 0y = 40
This simplifies to:
15x = 40
To solve for x, divide both sides of the equation by 15:
x = 40/15
Simplifying further:
x = 8/3 or approximately 2.67
Now, substitute the value of x back into either of the original equations to solve for y. Let's use the first equation, 5x + y = 24:
5(8/3) + y = 24
Simplifying:
40/3 + y = 24
Subtract 40/3 from both sides of the equation:
y = 24 - 40/3
To get a common denominator, multiply 24 by 3/3:
y = 24(3/3) - 40/3
Simplifying further:
y = 72/3 - 40/3
Subtracting the fractions:
y = 32/3 or approximately 10.67
So, the solution to the system of equations is x = 8/3 (approximately 2.67) and y = 32/3 (approximately 10.67).
To solve this system of equations using elimination, we want to eliminate one of the variables by multiplying one or both of the equations by suitable factors such that the coefficients of one of the variables will cancel out.
First, we can start by eliminating the x variable. We will do this by multiplying the first equation by 5 and the second equation by 1, which will give us:
(5)*(5x + y) = (5)*(24) ⇒ 25x + 5y = 120 (Equation 1)
(1)*(-5x - 4y) = (1)*(-56) ⇒ -5x - 4y = -56 (Equation 2)
Now, we can add Equation 1 and Equation 2 together:
(25x + 5y) + (-5x - 4y) = 120 + (-56)
25x + 5y - 5x - 4y = 64
(25x - 5x) + (5y - 4y) = 64
20x + y = 64 (Equation 3)
We have now simplified the system of equations to:
20x + y = 64 (Equation 3)
-5x - 4y = -56 (Equation 2)
Now we have a system of two equations with the same variables, x and y. We will solve this new system of equations using either substitution or elimination.
In this case, we will use elimination again to eliminate the y variable.
To eliminate y, we will multiply Equation 3 by 4 and Equation 2 by 5, which gives us:
(4)*(20x + y) = (4)*(64) ⇒ 80x + 4y = 256 (Equation 4)
(5)*(-5x - 4y) = (5)*(-56) ⇒ -25x - 20y = -280 (Equation 5)
Now, we add Equation 4 and Equation 5 together:
(80x + 4y) + (-25x - 20y) = 256 + (-280)
80x - 25x + 4y - 20y = -24
(80x - 25x) + (4y - 20y) = -24
55x - 16y = -24 (Equation 6)
We now have a new system of equations:
55x - 16y = -24 (Equation 6)
-5x - 4y = -56 (Equation 2)
Next, we can multiply Equation 2 by 11 to make the coefficients of x in both equations equal:
(11)*(-5x - 4y) = (11)*(-56)
-55x - 44y = -616 (Equation 7)
Now, we will add Equation 6 and Equation 7 together:
(55x - 16y) + (-55x - 44y) = -24 + (-616)
55x - 55x - 16y - 44y = -640
(-16y - 44y) = -640
-60y = -640
To solve for y, divide both sides of the equation by -60:
-60y/-60 = -640/-60
y = 10.67
Now that we have the value of y, we can substitute it back into one of the original equations (Equation 1, Equation 2, or Equation 3) to solve for x.
Using Equation 2:
-5x - 4(10.67) = -56
-5x - 42.68 = -56
-5x = -56 + 42.68
-5x = -13.32
Divide both sides of the equation by -5 to solve for x:
-5x/-5 = -13.32/-5
x = 2.664
Therefore, the solution to the system of equations is x ≈ 2.664 and y ≈ 10.67.