Please take me through this one problem and I think I can use this as a guide to tackle the others...
Solve by the addition method, indicate if it is independent, inconsistent, or dependent
x+3(y-1)=11
2(x-y)+8y=28
multiply out and get each into the form of ax + by = c.
then proceed. I will be happy to critique your work.
To solve this problem using the addition method, also known as the elimination method, we need to eliminate one of the variables by adding or subtracting the equations together. Here's how you can solve this problem step by step:
Step 1: Rearrange the equations in the form of ax + by = c.
The given equations are:
x + 3(y - 1) = 11 (Equation 1)
2(x - y) + 8y = 28 (Equation 2)
Start by simplifying Equation 1 by distributing 3 to y - 1:
x + 3y - 3 = 11
Simplify Equation 2 by distributing 2 to x - y:
2x - 2y + 8y = 28
Combine like terms:
x + 3y - 3 = 11
2x + 6y = 28
Step 2: Multiply one or both equations by appropriate constant(s) to make the coefficients of one of the variables equal or opposite.
In this case, you can multiply Equation 1 by 2 to make the coefficients of x in both equations equal:
2(x + 3y - 3) = 2(11)
2x + 6y - 6 = 22
Now we have the following set of equations:
2x + 6y - 6 = 22 (Equation 3)
2x + 6y = 28 (Equation 4)
Step 3: Subtract one equation from the other to eliminate the common variable.
Subtract Equation 3 from Equation 4:
(2x + 6y) - (2x + 6y - 6) = 28 - 22
Simplify by canceling out the common terms:
2x + 6y - 2x - 6y + 6 = 6
Combine like terms:
6 = 6
Step 4: Analyze the result.
When subtracting one equation from the other, we obtain an identity where 6 is equal to 6. This means that the equations are dependent, meaning they represent the same line, and have an infinite number of solutions. Any value of x and y that satisfies one equation will also satisfy the other.
In summary, the equations are dependent, indicating that they represent the same line and have an infinite number of solutions.