Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary. See Examples 3 and 4. 5x - 5y - 3 = 0 x - y - 12 = 0

Verified step by step guidance

1. Rewrite each equation in standard form, Ax + By = C. For the first equation, 5x - 5y - 3 = 0, rearrange to get 5x - 5y = 3. For the second equation, x - y - 12 = 0, rearrange to get x - y = 12.
2. Compare the coefficients of x and y in both equations. Notice that the coefficients in the second equation are exactly 1/5 of those in the first equation, suggesting the equations might be multiples of each other.
3. Substitute x - y = 12 into the first equation in place of x - y. This will test if the equations are indeed consistent or if they contradict each other.
4. Simplify the substitution to see if a true statement is formed, indicating the system has infinitely many solutions, or a false statement, indicating the system is inconsistent.
5. If the system has infinitely many solutions, express y in terms of x (or vice versa) using the equation x - y = 12, and state that x can be any arbitrary value.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Solutions can be unique, none, or infinitely many, depending on the relationships between the equations.

Inconsistent Systems

An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines that never intersect. In such cases, the equations contradict each other, indicating that there is no set of values for the variables that can satisfy all equations in the system.

Infinitely Many Solutions

A system of equations has infinitely many solutions when the equations represent the same line, meaning they overlap completely. This situation arises when one equation can be derived from the other through algebraic manipulation. The solution set can be expressed with one variable as arbitrary, indicating that there are countless combinations of values that satisfy the equations.