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. 2x = 3y + 4 4x = 3 - 5y

Verified step by step guidance

1. Rewrite both equations in a standard form to make them easier to work with. For the first equation, $\mathrm{2x}=\mathrm{3y}+\; 4$, subtract $3y$ and 4 from both sides to get $2x\; -\; 3y\; =\; 4$. For the second equation, $4x\; =\; 3\; -\; 5y$, add $5y$ to both sides and subtract 3 to get $4x\; +\; 5y\; =\; 3$.
2. Choose a method to solve the system: substitution or elimination. Here, elimination might be efficient. Multiply the first equation by 2 to align the coefficients of $x$: $2(2x\; -\; 3y)\; =\; 2(4)$ which simplifies to $4x\; -\; 6y\; =\; 8$.
3. Now subtract the second equation $4x\; +\; 5y\; =\; 3$ from the new equation $4x\; -\; 6y\; =\; 8$ to eliminate $x$. This gives $(4x\; -\; 6y)\; -\; (4x\; +\; 5y)\; =\; 8\; -\; 3$, simplifying to $-11y\; =\; 5$.
4. Solve for $y$ by dividing both sides by $-11$: $y\; =\; \backslash frac\{5\}\{-11\}\; =\; -\backslash frac\{5\}\{11\}$.
5. Substitute the value of $y$ back into one of the original equations, for example $2x\; -\; 3y\; =\; 4$, to solve for $x$. Replace $y$ with $-\backslash frac\{5\}\{11\}$ and solve the resulting equation for $x$.

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. 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 involves analyzing the equations' relationships, such as parallel lines for no solution or coincident lines for infinite solutions.

Set Notation for Solution Sets

Set notation expresses the solution set clearly and concisely. For example, a single solution is written as {(x, y)}, no solution as the empty set ∅, and infinitely many solutions as a set describing all points satisfying the system, often using a parameter.