solve the linear system using substitution
4x - y=5
4x + 4y= -4
Also by using elimination.
substitution: since you have y = 4x-5,
4x + 4(4x-5) = -4
20x = 16
x = 4/5
y = -9/5
using elimination, subtract one equation from the other to get
5y = -9
y = -9/5
now solve for x to get x=4/5
To solve the linear system using substitution, follow these steps:
Step 1: Solve one of the equations for one variable in terms of the other variable.
In this case, let's solve the first equation for x:
4x - y = 5
4x = y + 5
x = (y + 5)/4
Step 2: Substitute the expression for x into the other equation.
Now, substitute the value of x into the second equation:
4x + 4y = -4
4((y + 5)/4) + 4y = -4
(y + 5) + 4y = -4
5y + 5 = -4
5y = -9
y = -9/5
Step 3: Substitute the value of y back into any of the original equations to find the value of x.
Using the first equation:
4x - y = 5
4x - (-9/5) = 5
4x + 9/5 = 5
4x = 5 - 9/5
4x = 25/5 - 9/5
4x = 16/5
x = (16/5) / 4
x = 16/20
x = 4/5
Therefore, the solution to the linear system using substitution is x = 4/5 and y = -9/5.
To solve the linear system using elimination, follow these steps:
Step 1: Multiply one or both equations by a constant so that when added or subtracted, one of the variables eliminates.
In this case, let's multiply the first equation by 4:
4(4x - y) = 4(5)
16x - 4y = 20
Step 2: Add or subtract the two equations to eliminate one variable.
(16x - 4y) + (4x + 4y) = 20 + (-4)
20x = 16
x = 16/20
x = 4/5
Step 3: Substitute the value of x back into one of the original equations to solve for the remaining variable.
Using the first equation:
4x - y = 5
4(4/5) - y = 5
16/5 - y = 5
y = 16/5 - 25/5
y = (16 - 25)/5
y = -9/5
Therefore, the solution to the linear system using elimination is x = 4/5 and y = -9/5, which is the same as the solution found using substitution.