Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank(s). A_____of a system of equations is an ordered pair that satisfies each equation in the system.

Step-by-Step Solution

Answer

Solution

Step 1: Define the term

An ordered pair that satisfies every equation in a system simultaneously is called a specific term.

Step 2: Answer

The blank is filled with \(\textbf{solution}\).

Key Concepts

Solution

In mathematics, a solution refers to a set of values that satisfy a given condition or set of conditions. For a system of equations, the solution is the point at which all equations in the system intersect or hold true. This means finding the values of the variables that make each equation in the system accurate when plugged back in.
For example, in a system with two equations, the solution would be the specific values of the variables that satisfy both equations simultaneously.
Why is it important?

• Verifies the accuracy of your system of equations.
• Provides a common ground where all equations agree with each other.

By solving a system of equations, you are essentially finding where the different mathematical expressions meet and agree. A proper solution guarantees that each equation in the set is happy with the values it has received, ultimately leading to one consistent answer.

Ordered Pair

An ordered pair typically represents a solution to a system of two equations in two variables. This pair consists of two elements written in a specific order \(x, y\) and describes a point in a two-dimensional space, usually on a coordinate plane. These are usually values for \(x\) and \(y\), and the order is crucial because \(x\) corresponds to a position on the horizontal axis, while \(y\) corresponds to a position on the vertical axis.
The logic behind "ordered" is important because switching these numbers would place the point in a completely different location.
Consider:

• (3, 5) indicates moving 3 units to the right and 5 units up from the origin on a standard plane.
• Reversing to (5, 3) would instead mean moving 5 units right and just 3 units up, placing the point elsewhere.

Thus, each ordered pair uniquely identifies a location and solution where this pair suits the equations of a system.

Satisfies Equations

To say an ordered pair satisfies a system of equations means that when you substitute the pair into each equation, the equations remain true. In other words, plugging the values of the ordered pair into each equation should solve the equations perfectly with no leftover or unbalanced terms.
Much like a test to determine the correctness of a solution, you are confirming by checking:

• If the equation \(x + y = 8\) is satisfied by \(\(x, y\) = (5, 3)\), substituting gives \(5 + 3 = 8\), which is true.
• Checking it against another equation like \(y - 2x = -2\) with \(x = 5\) and \(y = 3\), we substitute to get \(3 - 2(5) = -7\), which is not satisfied.

If a pair fails to satisfy an equation in the system, it is not a valid solution. Therefore, a solution must satisfy all equations in the system simultaneously, ensuring that the "satisfaction" of each equation is perfectly achieved with that ordered pair.