Problem 6 - College Algebra

Question

Determine whether the given ordered pair is a solution to the system of equations. $$ \begin{array}{l} 5 x-y=4 \\ x+6 y=2 \text { and }(4,0) \end{array} $$

Step-by-Step Solution

Answer

The ordered pair \((4, 0)\) is not a solution to the system of equations.

Step 1: Identify the variables in the ordered pair

The ordered pair given is \((4, 0)\). Here, \(x = 4\) and \(y = 0\). We will substitute these values into both equations of the system.

Step 2: Substitute into the first equation

Substitute \(x = 4\) and \(y = 0\) into the first equation \(5x - y = 4\). This gives us \(5(4) - 0 = 20\). Since \(20 eq 4\), the ordered pair \((4, 0)\) does not satisfy the first equation.

Step 3: Substitute into the second equation

Now, substitute \(x = 4\) and \(y = 0\) into the second equation \(x + 6y = 2\). This results in \(4 + 6(0) = 4\). Since \(4 eq 2\), the ordered pair \((4, 0)\) also does not satisfy the second equation.

Step 4: Conclusion on whether it is a solution

Since the ordered pair \((4, 0)\) does not satisfy either equation in the system, it is not a solution to the system of equations.

Key Concepts

Understanding Ordered Pairs

In mathematics, an *ordered pair* is a way to represent two elements that are related, typically in a two-dimensional space. It is written in the form \((x, y)\), where 'x' is the first element and 'y' is the second. Ordered pairs are often used in relation to graphs, where they indicate a point's position on the Cartesian plane.

• The first number (x-coordinate) shows the position along the horizontal axis (left-right).
• The second number (y-coordinate) shows the position along the vertical axis (up-down).

When dealing with systems of equations, ordered pairs are crucial as they represent potential solutions to the equations involved.
In our exercise, the ordered pair given is \((4, 0)\), which specifies thex value as 4 and they value as 0. By substituting these values into each equation in the system, we determine if the pair is a solution to the equations.

Exploring the Substitution Method

The *substitution method* is one of the techniques used to solve systems of equations. This method involves solving one of the equations for one variable in terms of another and then substituting this expression into the other equation.
Here's how it typically works:

• First, solve one of the equations for one variable.
• Then, substitute that expression into the other equation.
• Solve the resulting equation for the other variable.
• Finally, substitute back to find the first variable.

In the exercise at hand, instead of solving, we're checking if a specific ordered pair \((4, 0)\) fits as a solution. Thus, we substitute \(x = 4\) and \(y = 0\) into each equation to see if both sides of the equations remain true, which we found they did not.

Determining the Solution to Equations

A *solution to a system of equations* is any set of values for the variables that make all equations in the system true simultaneously. In the case of two-variable systems like the one in our exercise, solutions are represented as ordered pairs.
To check if an ordered pair is a solution:

• Substitute the values from the ordered pair into each equation of the system.
• If the substituted values make both equations true (the left and right side equal), then the ordered pair is a solution.
• If even one equation is false after substitution, the pair is not a solution to the system.

For example, in our exercise, the ordered pair \((4, 0)\) was tested. Neither equation became true with these substitutions, so this pair is not a solution for the given system of equations. Understanding this process is crucial for solving or verifying solutions to systems of equations effectively.