In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. y = 3x - 5 21x - 35 = 7y

Verified step by step guidance

1. Start by writing down the given system of equations: $y=3x-5$ and $21x\; -\; 35\; =\; 7y$.
2. Substitute the expression for $y$ from the first equation into the second equation to eliminate $y$. This means replacing $y$ in $21x\; -\; 35\; =\; 7y$ with $3x\; -\; 5$.
3. After substitution, simplify the resulting equation: $21x\; -\; 35\; =\; 7(3x\; -\; 5)$. Use the distributive property to expand the right side.
4. Compare both sides of the simplified equation. If the variables cancel out and you get a true statement (like $0\; =\; 0$), the system has infinitely many solutions. If you get a false statement (like $0\; =\; 5$), the system has no solution.
5. If the equation simplifies to a true statement, express the solution set using set notation with the parameter $x$, such as $\{(x,\; y)\; |\; y\; =\; 3x\; -\; 5,\; x\; \; \}$. If it has no solution, state that the solution set is the empty set $$.

Key Concepts

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding all variable values that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's form.

Identifying No Solution and Infinite Solutions

A system has no solution if the equations represent parallel lines that never intersect, indicating inconsistency. It has infinitely many solutions if the equations represent the same line, meaning all points on the line satisfy both equations. Recognizing these cases involves comparing slopes and intercepts or simplifying equations.

Expressing Solution Sets Using Set Notation

Set notation is a concise way to describe all solutions of a system. For infinite solutions, it often involves a parameter representing all points on a line, e.g., {(x, y) | y = 3x - 5}. For no solution, the solution set is the empty set, denoted by ∅, indicating no ordered pairs satisfy the system.