Problem 2 - Elementary Algebra Within Reach

Question

In Exercises \(1-4\), determine whether each ordered pair is a solution of the system. $$ \left\\{\begin{aligned} 3 x-y &=-2 \\ x-3 y &=2 \end{aligned}\right. $$ (a) \((0,2)\) (b) \((-1,-1)\)

Step-by-Step Solution

Answer

The ordered pair (0,2) is not a solution to the system of equations, but the ordered pair (-1,-1) is.

Step 1: Check the ordered pair (0,2)

First of all, substitute \(x = 0\) and \(y = 2\) into the two equations. The first equation becomes \(3*0 - 2 = -2\) which simplifies to \(-2 = -2\). This is true. The second equation becomes \(0 - 3*2 = 2\), which simplifies to \(-6 = 2\). This is not true. Therefore, the ordered pair (0,2) is not a solution to the system of equations.

Step 2: Check the ordered pair (-1,-1)

Now, substitute \(x = -1\) and \(y = -1\) into the two equations. The first equation becomes \(3*(-1) - (-1) = -2\), which simplifies to \(-3 + 1 = -2\), or \(-2 = -2\). This is true. The second equation becomes \(-1 - 3*(-1) = 2\), which simplifies to \(-1 + 3 = 2\), or \(2 = 2\). This is also true. Therefore, the ordered pair (-1,-1) is a solution to the system of equations.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially when dealing with systems of equations. An ordered pair consists of two elements written in a specific order, commonly represented as

• \((x, y)\),

where \(x\) corresponds to the horizontal coordinate, and \(y\) corresponds to the vertical coordinate. These pairs are essential in plotting points on the Cartesian plane, where

• \(x\) determines the position along the horizontal axis,
• \(y\) determines the position along the vertical axis.

For instance, the ordered pair (0,2) means moving 0 units horizontally and 2 units vertically from the origin (0,0). In solving systems of equations, ordered pairs serve as potential solutions that satisfy both equations simultaneously. The goal is to find the pair \((x, y)\) that makes both equations true.

Solution of a System of Equations

A system of equations is a set of two or more equations with the same variables. Finding the solution to a system of equations means identifying the ordered pair that satisfies all the equations in the system. Visually, this represents the point where all the expression's graphs intersect in a coordinate plane.

In the problem at hand, the system of equations is:

• \[ \begin{aligned} 3x - y &= -2, \ x - 3y &= 2. \end{aligned} \]

To check if a specific ordered pair, like

• (0,2)
• (-1,-1),

is a solution, each pair is substituted into the equations. If both equations are true after substitution, then the ordered pair is indeed the solution to the system.

From the solution process, we see that only the ordered pair (-1,-1) satisfies both equations, confirming it as the solution to the system. This implies that the graphs of these equations will intersect at this point on a graph.

Substitution Method Explained

The substitution method is an effective technique for solving systems of equations. This method involves solving one of the equations for one variable, then substituting that expression into the other equation. Through this approach, the system reduces to a single equation in one variable, which simplifies the solution process.

Here's the general process:

• Select one of the equations and solve it for one variable.
• Substitute this expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute back to find the other variable, if necessary.

In our problem, however, we checked specific ordered pairs rather than deriving them through substitution. If we were to apply substitution, we might solve one equation for \(x\) or \(y\), and substitute into the other to check the solution analytically. This method ensures precision, especially when verifying or deriving solutions systematically.