I'm confused regarding the elimination method for
5x - 7y = -16,
2x + 8y = 26
5 x - 7 y = - 16 Multiply both sides by - 2
- 10 x + 14 y = 32
2 x + 8 y = 26 Multiply both sides by 5
10 x + 40 y = 130
- 10 x + 14 y = 32
10 x + 40 y = 130 Add equations
- 10 x + 10 x + 14 y + 40 y = 32 + 130
54 y = 162 Divide both sides by 54
y = 162 / 54 = 3
5 x - 7 y = - 16
5 x - 7 * 3 = - 16
5 x - 21 = - 16 Add 21 to both sides
5 x - 21 + 21 = - 16 + 21
5 x = 5 Divide both sides by 5
x = 1
Solutions :
x = 1 , y = 3
#1: 5x - 7y = -16
#2: 2x + 8y = 26
To eliminate y, multiply #1 by 8 and #2 by 7 to get
40x - 56y = -128
14x + 56y = 182
If you add those two new equations, the y's are eliminated and you get
54x = 54
x=1
Now you can go back and calculate y from one of the original equations.
To solve the given system of equations using the elimination method, follow these steps:
Step 1: Multiply the first equation by 2 and the second equation by 5 to make the coefficients of x in both equations the same value.
Multiplying the first equation by 2:
2 * (5x - 7y) = 2 * (-16)
10x - 14y = -32
Multiplying the second equation by 5:
5 * (2x + 8y) = 5 * 26
10x + 40y = 130
Step 2: Add the equations together to eliminate the x term.
(10x - 14y) + (10x + 40y) = -32 + 130
10x + 10x - 14y + 40y = 98
Combining like terms:
20x + 26y = 98
Step 3: Simplify the equation obtained in Step 2.
Divide the equation by 2:
(20x + 26y)/2 = 98/2
10x + 13y = 49
Now, you have a new equation:
10x + 13y = 49
Step 4: Solve the new equation for one variable.
Let's solve for x:
10x + 13y = 49
10x = 49 - 13y
x = (49 - 13y)/10
Step 5: Substitute the x-value obtained into one of the original equations to find the corresponding value of y.
Using the first equation:
5x - 7y = -16
5((49 - 13y)/10) - 7y = -16
Simplify:
(245 - 65y)/10 - 7y = -16
Multiply through by 10 to clear the fraction:
245 - 65y - 70y = -160
Combine like terms:
-135y = -405
Divide by -135:
y = -405/-135
y = 3
Step 6: Substitute the y-value back into either of the original equations to find the corresponding value of x.
Using the second equation:
2x + 8y = 26
2x + 8(3) = 26
2x + 24 = 26
2x = 26 - 24
2x = 2
x = 1
Therefore, the solution to the system of equations is x = 1 and y = 3.
To solve the given system of equations using the elimination method, follow these steps:
Step 1: Multiply the equations by suitable numbers to make the coefficients of one of the variables opposites in the two equations. In this case, let's choose to eliminate the variable "y." Looking at the coefficients of "y" in both equations, we can multiply the first equation by 8 and the second equation by -7.
8*(5x - 7y) = 8*(-16) [Multiply the first equation by 8]
-7*(2x + 8y) = -7*(26) [Multiply the second equation by -7]
This will give us:
40x - 56y = -128
-14x - 56y = -182
Step 2: Add the two equations obtained in step 1. By doing this, we eliminate the variable "y."
(40x - 56y) + (-14x - 56y) = -128 + (-182)
40x - 56y - 14x - 56y = -310
(40x - 14x) + (-56y - 56y) = -310
26x - 112y = -310
So, the new equation after adding both equations is: 26x - 112y = -310.
Step 3: Solve the new equation obtained in step 2 for the remaining variable. In this case, let's solve for "x."
26x - 112y = -310
To isolate the "x" term, we can move the "-112y" term to the right side of the equation by adding 112y to both sides:
26x = 112y - 310
Next, divide both sides of the equation by 26 to solve for "x":
x = (112y - 310)/26
Step 4: Substitute the value of "x" obtained in step 3 into either of the original equations to solve for "y." Let's substitute it into the first equation:
5x - 7y = -16
5((112y - 310)/26) - 7y = -16
Now simplify and solve the equation for "y."