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Chapter 7: Problem 1

Determine whether the ordered pair (-2,1) is a solution to each equation. $$x+y=-1$$

Short Answer

Expert verified
Yes, the ordered pair (-2, 1) is a solution to the equation \(x+y=-1\).

Step by step solution

01

Identify the coordinates

The given pair (-2, 1) represents the values of x and y. It means x is -2 and y is 1.
02

Substitute the values of x and y in the equation

Now place these numbers into the equation \(x+y=-1\), and replace x with -2 and y with 1. The equation then becomes \(-2+1=-1\).
03

Verify if both sides of the equation are equal

Solving the equation \(-2 + 1\), we get \(-1\). Therefore, \(-1 = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
An ordered pair consists of two elements鈥攊n the context of Cartesian coordinates, they represent the x (horizontal) and y (vertical) coordinates of a point on a plane. The ordered pair (a, b) communicates the position of a point uniquely: 'a' pertains to the point's location along the x-axis, while 'b' represents the y-axis location.

For example, the ordered pair (-2,1) points us to a specific location on the Cartesian plane: two units to the left of the origin on the x-axis and one unit above the origin on the y-axis. The fact that the pair of numbers is 'ordered' is crucial; if reversed, the position would change completely, taking us to a different point on the graph.
Substitution Method
The substitution method is a staple in solving algebraic equations, particularly systems of equations. It involves replacing variables with their corresponding values to simplify an equation or determine the validity of potential solutions.

In practice, if given an ordered pair that might be a solution to an equation, you effectively 'substitute' the first number of the pair for x and the second for y. Then, you perform the necessary arithmetic operations to see if the equation holds true. This method is especially helpful when dealing with linear equations, allowing you to determine whether an ordered pair is part of the equation's graph.
Solution Verification
Solution verification is the process of confirming whether a proposed solution satisfies a given equation. After substituting the potential solution into the equation, as shown in our example with the ordered pair (-2,1), it either confirms or rejects the solution based on the equality of both sides of the equation.

To verify a solution effectively, every step of the substitution must be carried out with attention to detail, ensuring that arithmetic operations are accurately performed. This attention to detail ensures that we don't incorrectly validate or discard a potential solution. It's a critical final check in problem-solving that underscores accurate and reliable results.
Linear Equations
Linear equations are fundamental in algebra and are characterized by variables to the first power and a constant. They are graphically represented as straight lines when plotted on a coordinate plane. The general form of a linear equation is given as y = mx + b, where m is the slope, and b is the y-intercept.

In terms of solutions, a linear equation will have an infinite number of ordered pairs that satisfy it, each corresponding to a point on the line it represents. Understanding linear equations is pivotal because they form the basis for more complex algebraic concepts, modeling real-world situations with a clear and direct relationship between variables.

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