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. x/4 - y/4 = −1 x + 4y = -9

Verified step by step guidance

1. Step 1: The first step is to simplify the equations. The first equation can be simplified by multiplying both sides by 4 to get rid of the fractions. This will give us x - y = -4.
2. Step 2: Now we have two equations: x - y = -4 and x + 4y = -9. We can solve this system of equations using either substitution or elimination method. Let's use the elimination method. To do this, we need to make the coefficients of y the same in both equations. We can multiply the first equation by 4 to achieve this. This gives us 4x - 4y = -16.
3. Step 3: Now we have two new equations: 4x - 4y = -16 and x + 4y = -9. We can add these two equations together. The y terms will cancel out, leaving us with an equation in terms of x.
4. Step 4: Solve the resulting equation from step 3 for x.
5. Step 5: Substitute the value of x obtained in step 4 into one of the original equations to find the value of y.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. Systems can be classified as consistent (having at least one solution) or inconsistent (having no solutions). Understanding how to manipulate and analyze these equations is crucial for finding solutions.

Types of Solutions

In the context of systems of equations, solutions can be classified into three categories: a unique solution, no solution, and infinitely many solutions. A unique solution occurs when the lines represented by the equations intersect at a single point. No solution arises when the lines are parallel, while infinitely many solutions occur when the equations represent the same line, leading to an infinite number of intersection points.

Set Notation

Set notation is a mathematical way to describe a collection of objects, often used to express the solution sets of equations. For example, a unique solution can be expressed as a single ordered pair (x, y), while infinitely many solutions may be represented using parameterization or interval notation. Understanding how to use set notation is essential for clearly communicating the nature of the solutions found in a system of equations.