Which method would be the simplest way to solve the system
7x+5y=19
-7x-2y=-16
The simplest method to solve this system of equations would be to use the method of addition or elimination.
To do this, we can add both equations together so that the x-terms cancel out.
(7x + 5y) + (-7x - 2y) = 19 + (-16)
5y - 2y = 3
3y = 3
y = 1
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x.
7x + 5(1) = 19
7x + 5 = 19
7x = 14
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 1.
To solve the given system of equations:
1. Multiply the second equation by -1 to eliminate the x term:
-1(-7x - 2y) = -1(-16)
7x + 2y = 16
2. Now, add the two equations together to eliminate the x terms:
(7x + 5y) + (7x + 2y) = 19 + 16
14x + 7y = 35
3. Divide the equation by 7 to simplify:
(14x + 7y) / 7 = 35 / 7
2x + y = 5
4. Now, solve the simplified equation for y in terms of x:
y = 5 - 2x
5. Substitute the value of y into one of the original equations (preferably the first equation):
7x + 5(5 - 2x) = 19
6. Solve for x:
7x + 25 - 10x = 19
-3x = -6
x = 2
7. Substitute the value of x into the equation derived in step 4:
y = 5 - 2(2)
y = 5 - 4
y = 1
8. Therefore, the solution to the system of equations is x = 2 and y = 1.
To solve the system of equations:
7x + 5y = 19 ...(Equation 1)
-7x - 2y = -16 ...(Equation 2)
One of the simplest methods to solve this system of equations is by using the elimination method. Here's how you can proceed:
Step 1: Multiply the second equation by 5 to make the coefficients of 'x' in both equations add up to zero when added together.
-7x - 2y = -16
Multiply by 5 on both sides:
(-7x - 2y) * 5 = (-16) * 5
-35x - 10y = -80 ...(Equation 3)
Step 2: Add Equation 1 and Equation 3 together to eliminate 'x':
(7x + 5y) + (-35x - 10y) = 19 + (-80)
Combine like terms:
7x - 35x + 5y - 10y = 19 - 80
-28x - 5y = -61 ...(Equation 4)
Step 3: Multiply Equation 4 by (-1) to make the coefficient of 'x' positive:
(-1) * (-28x - 5y) = (-1) * (-61)
28x + 5y = 61 ...(Equation 5)
Step 4: Add Equation 4 and Equation 5 together to eliminate 'y':
(-28x - 5y) + (28x + 5y) = (-61) + (61)
Combine like terms:
-28x + 28x - 5y + 5y = 0
0 = 0
Step 5: The equation 0 = 0 indicates that the two equations are dependent, meaning there are infinitely many solutions. This means that the system of equations is consistent and has dependent equations.
Therefore, the simplest way to solve this system of equations is to recognize that they are dependent, meaning there are infinitely many solutions.