Problem 36 - College Algebra

Question

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. \(2 x+5 y=-4\) \(3 x-y=11\)

Step-by-Step Solution

Answer

The solutions to the system of equations are x = 3 and y = 0. This system has a unique solution.

Step 1: Arrange the Equations

Firstly, arrange both equations in such a manner that it is easy to handle. Arrange both equations like this: \(2x + 5y = -4\) and \(3x - y = 11\)

Step 2: Use Elimination Method

To use the elimination method, we multiply the second equation by 5 so that we can cancel out y when the equations are combined. So second equation becomes \(15x - 5y = 55\).

Step 3: Combine Both Equations

Combine both of the equations. The equation becomes \(2x + 5y + 15x - 5y = -4 + 55\). Solving this we get the value of x: \(17x = 51\), or \(x = 3\).

Step 4: Substituting x in Second Equation

Substitute x in second equation:\(3x - y = 11\). Replace \(x\) with \(3\) we get \(y = 0\).

Step 5: Conclusion

After solving the system of equations, we have a unique solution. The solution of the system is \((x,y) = (3,0)\). There are no other solutions for this system.

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It involves combining the equations in such a way that one of the variables is eliminated, making it easier to solve for the remaining variable. This method is especially useful when the coefficients of one of the variables can be made equal by multiplication.
To use elimination effectively:

• Align the given equations properly.
• Multiply one or both equations to align coefficients for elimination.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting single-variable equation.

Once one variable is found, substitute it back into one of the original equations to find the other variable. The practice of elimination can make solving complex systems more streamlined.

Unique Solution

A unique solution in the context of a system of equations refers to a single set of values for the variables that satisfy all the equations in the system. In other words, there's only one point where all the equations intersect. For a system of linear equations, this occurs when the equations are independent and consistent.
To determine if a system has a unique solution:

• Ensure the equations are not multiples of each other.
• Perform algebraic manipulations to look for contradictions or redundancies.
• If no contradictions exist, and the system reduces to specific values, it has a unique solution.

In graphical terms, this means the lines represented by the equations intersect at exactly one point.

Set Notation

Set notation is a mathematical method used to express a collection of objects or numbers clearly and concisely. It is often used in the context of solutions to denote the elements that satisfy given conditions.
For systems of equations:

• Solutions with no intersection are often written as an empty set, symbolized by \( \emptyset \).
• If there's a unique solution, it can be written as a set containing that point, such as \( \{(3, 0)\} \).
• For infinitely many solutions, a general form can be expressed in set notation.

Set notation helps standardize the way we communicate solutions, making it easier to understand and compare different solution strategies.

Solutions of Linear Equations

Solutions of linear equations are the values of the variables that make all the equations in a system true simultaneously. Depending on the relationships between the equations, the solutions can vary significantly.
Here are the typical types of solutions:

• Unique Solution: One specific pair or set of values that satisfies all equations.
• No Solution: No set of values can simultaneously satisfy all equations, often due to parallel lines.
• Infinitely Many Solutions: The equations are dependent, leading to overlapping lines and multiple solutions.

The process of finding solutions involves analyzing the equations and using methods like elimination or substitution to discover the values that work. Each type of solution tells us something different about the nature of the equations and their relationships.