Solve each system by the substitution method. First simplify each equation by combining like ter

step 1 We need to explain how to identify a system with no solution when we solve with substitution. So

Solve each system by the substitution method. First simplify...

Key Concepts

Identifying Inconsistent Systems

An inconsistent system is one that has no solution, often revealed during the substitution process when the manipulation of the equations leads to a false statement, such as 0 = 5. This indicates that the equations represent parallel lines that never intersect. Recognizing such a contradiction during the substitution method helps to determine early on that the system does not have any common solution.

Combining Like Terms

Combining like terms means to simplify algebraic expressions by summing coefficients of terms that have the same variable part. This process is essential in simplifying equations before applying the substitution method, as it ensures that the equations are in their simplest form, making subsequent calculations clearer and reducing the potential for errors.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others, and then substituting that expression into the remaining equation(s). This technique transforms a system of equations into one that is easier to solve, typically reducing the number of variables in an equation and facilitating the finding of solutions.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal when solving a system is to find values for the variables that satisfy all the equations simultaneously. In the context of linear equations, this commonly involves finding the intersection point of lines, which represents the unique solution when one exists.

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