Solve this system 10x+2y=7,y=-5x+3 by addition or substitution. If a unique solution does not exist state if the system is dependent or inconsistent.
Since the second equation can be rewritten 5x + y = 3, or 10x + 2y = 6, it is inconsistent with the first equation, which says that 10x + 2y = 7. Therefore there is no solution.
To solve the given system of equations using either addition or substitution method, we will follow these steps:
Using the Substitution Method:
1. Start with the second equation: y = -5x + 3.
2. Substitute this expression for y into the first equation: 10x + 2(-5x + 3) = 7.
3. Simplify the equation by distributing the 2: 10x - 10x + 6 = 7.
4. Combine like terms: 6 = 7.
5. Since the equation 6 = 7 is not true, this means the system is inconsistent, meaning there is no solution.
Using the Addition Method:
1. Multiply the second equation by 2 to make the coefficients of y the same: 2y = -10x + 6.
2. Rearrange the terms to write the equation in the standard form: 10x + 2y = 6.
3. Now, we have the system:
10x + 2y = 7 (Equation 1)
10x + 2y = 6 (Equation 2)
4. Subtract Equation 2 from Equation 1 to eliminate the x-variable:
(10x + 2y) - (10x + 2y) = 7 - 6.
This simplifies to 0 = 1, which is not true.
5. Since the equation 0 = 1 is not true, this means the system is inconsistent.
In conclusion, the given system of equations is inconsistent, meaning it does not have a unique solution.