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. 3x - 2y = − 5 4x + y = 8

Verified step by step guidance

1. Start by writing down the system of equations clearly: $\mathrm{3x}-2y=-5$ and $4x+y=8$.
2. Choose a method to solve the system. Here, substitution or elimination are both good options. For elimination, aim to eliminate one variable by making the coefficients of either $x$ or $y$ opposites.
3. Multiply the second equation by 2 to align the coefficients of $y$: $2(4x\; +\; y)\; =\; 2(8)$ which gives $8x\; +\; 2y\; =\; 16$.
4. Add the first equation $3x\; -\; 2y\; =\; -5$ and the new equation $8x\; +\; 2y\; =\; 16$ to eliminate $y$: $(3x\; +\; 8x)\; +\; (-2y\; +\; 2y)\; =\; -5\; +\; 16$.
5. Solve the resulting equation for $x$, then substitute this value back into one of the original equations to find $y$. Finally, check if the system has one solution, no solution, or infinitely many solutions by analyzing the consistency of the equations.

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 values for the variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing.

Types of Solutions for Systems

Systems of linear equations can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent). Identifying the type depends on the relationships between the equations, such as parallel lines or coincident lines.

Expressing Solution Sets Using Set Notation

Set notation is a concise way to represent the solution(s) of a system. For a unique solution, it lists the ordered pair; for infinitely many solutions, it expresses the solution in terms of a parameter; and for no solution, the solution set is the empty set, denoted by ∅.