how do i solve this linear system using substitution or elimination
-16x+2y= -2
y=8x-1
Substitution would be the best
-16x +2(8x-1)=-2
To solve the given linear system using substitution or elimination, we can follow these steps:
Method 1: Substitution
Step 1: Solve one equation for one variable (usually the equation that is already solved for one variable).
Let's solve the second equation for y:
y = 8x - 1
Step 2: Substitute the expression obtained for the variable into the other equation.
Substitute the value of y in the first equation:
-16x + 2(8x - 1) = -2
Step 3: Simplify and solve for the remaining variable.
Expand and simplify:
-16x + 16x - 2 = -2
-2 = -2
Step 4: Analyze the result.
In this case, both variables have canceled each other out, leaving only a true statement (-2 = -2). This tells us that the two equations represent the same line, meaning there are infinitely many solutions to the system.
Method 2: Elimination
Step 1: Multiply one or both equations by suitable constants to make the coefficients of one of the variables the same (or opposite) in both equations.
In this case, it's already convenient to eliminate the variable 'y' by multiplying the first equation by 4, and the second equation by 2:
-64x + 8y = -8 (equation 1)
16x - 2y = -2 (equation 2)
Step 2: Add or subtract the two equations to eliminate one of the variables.
Adding both equations:
(-64x + 8y) + (16x - 2y) = -8 + (-2)
-48x + 6y = -10
Step 3: Solve the resulting equation for the remaining variable.
Simplify and solve for x:
-48x + 6y = -10
-48x = -10 - 6y
x = (10 + 6y) / 48
x = (5 + 3y) / 24
Step 4: Substitute the solution for x into any of the original equations to solve for the other variable.
Let's substitute the value of x into the second equation:
y = 8x - 1
y = 8((5 + 3y) / 24) - 1
Simplify and solve for y:
y = (40 + 24y) / 24 - 1
24y = 40 + 24y - 24
24y - 24y = 40 - 24
0 = 16
Step 5: Analyze the result.
In this case, we obtained a false statement (0 = 16). This tells us that there is no solution to the system.
Therefore, the given linear system is inconsistent (has no solution).