Problem 3 - Elementary Algebra Within Reach

Question

In Exercises \(1-4\), determine whether each ordered pair is a solution of the system. $$ \left\\{\begin{array}{r} 2 x-3 y=-8 \\ x+y=1 \end{array}\right. $$ (a) \((5,-3)\) (b) \((-1,2)\)

Step-by-Step Solution

Answer

The ordered pair (5,-3) is a solution to the system, while the pair (-1,2) is not a solution.

Step 1: Substitute (5,-3) into the System

Replace every x in both equations with 5 and every y with -3. Then, simplify both equations. If both equations equal, then the ordered pair is a solution to the system.

Step 2: Substitute (-1,2) into the System

This time, replace every x with -1 and every y with 2 in both equations. Similar to step 1, simplify both equations. If both equations equal, then this ordered pair is also a solution to the system.

Step 3: Compare results

Compare the results from step 1 and 2 to the respective right sides of the equations. If they match, the pairs are solutions; otherwise, they are not.

Step 4: State final answer

After identifying whether the given ordered pairs are solutions or not for the system of equations, state the final answers.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one of the equations for one variable in terms of the other, and then substituting this expression into the second equation. This method is highly effective when one of the equations in the system is easy to solve for one of the variables.
For example, consider the system:

• \( 2x - 3y = -8 \)
• \( x + y = 1 \)

To use substitution, you might choose the second equation \( x + y = 1 \) and solve for \( x \):

• \( x = 1 - y \)

Now, substitute \( 1 - y \) for \( x \) in the first equation \( 2x - 3y = -8 \):

• \( 2(1 - y) - 3y = -8 \)
• Simplify to find \( y \).

This method helps to reduce the number of equations to solve, making the system manageable.

Ordered Pairs

Ordered pairs are a fundamental concept in coordinate geometry and systems of equations. An ordered pair is simply a pair of numbers used to locate a point in a coordinate system. It is written in the form \( (x, y) \), where \( x \) is the horizontal coordinate (often referred to as the abscissa) and \( y \) is the vertical coordinate (often called the ordinate).
In the context of systems of equations, an ordered pair represents a potential solution. It indicates the values for \( x \) and \( y \) that satisfy all equations in the system simultaneously.
To verify if a given ordered pair is indeed a solution of the system, substitute the \( x \) and \( y \) values into each equation of the system. If both equations hold true, the ordered pair is a solution.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together since they share common variables. The idea is to find the values of these variables that satisfy all the equations in the system at the same time.
Consider this system:

• \( 2x - 3y = -8 \)
• \( x + y = 1 \)

These equations are considered simultaneous because they need to be solved together. For an ordered pair to be a solution to this system, it must make both equations true simultaneously.
Solving simultaneous equations can be approached through various methods, like substitution or the elimination method. Choosing the right method depends on the structure of the equations for simplicity and efficiency.

Algebraic Solution

Finding an algebraic solution to a system of equations involves using algebraic techniques to find the values of the variables that satisfy all the equations. In this context, it means finding values for \( x \) and \( y \) such that both given equations hold true.
Using algebraic techniques often involves:

• Isolating variables by rearranging terms.
• Substituting expressions from one equation into another.
• Simplifying expressions through addition, subtraction, multiplication, or division.

The ultimate goal is to find a set of values — an ordered pair — that works for all the equations in the system. In simpler systems, like the provided ones, the algebraic solution can often be found by straightforward substitution or elimination. This approach provides a precise answer through logical mathematical reasoning.