Problem 1 - Thinking Mathematically

Question

In Exercises 1-4, determine whether the given ordered pair is a solution of the system. \((2,3)\) \(\left\\{\begin{array}{l}x+3 y=11 \\ x-5 y=-13\end{array}\right.\)

Step-by-Step Solution

Answer

Yes, the ordered pair (2,3) is a solution of the system of equations.

Step 1: Substitute the ordered pair into the first equation

We begin by substituting x = 2 and y = 3 into our first equation: \( x + 3y = 11 \). This gives us: \( 2 + 3*3 = 11 \). Simplifying this results in 11 = 11, which is true, so (2,3) satisfies the first equation.

Step 2: Substitute the ordered pair into the second equation

We now substitute x = 2 and y = 3 into our second equation: \( x - 5y = -13 \). This gives us: \( 2 - 5*3 = -13 \). Simplifying this results in -13 = -13, which is also true, so (2,3) satisfies the second equation as well.

Step 3: Determine if the ordered pair is a solution

Since substituting the ordered pair (2,3) into both equations results in true statements, we can conclude that (2,3) is indeed a solution to the given system of equations.

Key Concepts

Ordered Pair

An ordered pair is a way to represent a point in a two-dimensional space. It consists of two elements placed in a specific sequence, usually written in the form \(x, y\). The first element corresponds to a position along the horizontal axis (x-axis), and the second element corresponds to a position along the vertical axis (y-axis).

For example, the ordered pair \(2, 3\) means the 'x' value is 2, and the 'y' value is 3. In the context of solving systems of equations, an ordered pair provides specific values for the variables, allowing us to check if it is a solution to the equation or system of equations provided.

Solution Verification

Solution verification is the process of checking whether a particular ordered pair satisfies the equations in a system.

When we have an ordered pair, like \(2, 3\), we substitute it into each equation in the system to see if the equations hold true:

• Substituting 'x' and 'y' values from the ordered pair into the equations.
• Checking if the left and right sides of each equation produce equal results.

This process is crucial because if the equations are true for the provided values, then the ordered pair is indeed a solution to the system. As shown in the exercise, after substitution, both equations were satisfied, confirming that \(2, 3\) is a solution.

Substitution Method

The substitution method is one of the powerful techniques to solve systems of linear equations. It involves replacing one variable with an expression obtained from another equation.

Here's a simple rundown:

• First, solve one of the equations for one of the variables, such as expressing 'x' in terms of 'y' or vice versa.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation to find the value of one variable.
• Finally, use this value to find the other variable by substitution.

In the exercise, though substitution was used for checking, this method can be a strategy to find solutions initially if the ordered pair was unknown.

Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. These equations typically have variables raised only to the power of one. They often take the form \(ax + by = c\), where 'a,' 'b,' and 'c' are constants.

In the system of equations from the exercise, both \(x + 3y = 11\) and \(x - 5y = -13\) are linear equations. Each equation represents a line, and their intersection, if any, would be a point (an ordered pair) that satisfies both equations at once. That's why the process checks if the point \(2, 3\) lies on both lines, confirming it as a solution to the system.