Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

For an equation in \(x\) and \(y,\) if substitution of \(a\) for \(x\) and \(b\) for \(y\) satisfics the equation, then the point \((a, b)\) is a _____.

Step-by-Step Solution

Answer

The point \((a, b)\) is a solution.

Step 1: Understanding Terminology

In algebra, when we substitute values into an equation and the equation holds true (i.e., both sides of the equation are equal), we say the values satisfy the equation. So if the values of 'a' for 'x' and 'b' for 'y' satisfy an equation, then the point (a, b) is a solution to the equation.

Step 2: Relation to Coordinate Geometry

In coordinate geometry, a point that satisfies an equation is a point that lies on the graph of the equation in a coordinate plane. This means that the values of that point (coordinates) satisfy the equation of the graph.

Step 3: Classifying a Point on the Graph

Therefore, when we say that the point (a, b) satisfies the equation, we mean that this point lies on the graph of the equation. Hence, if substitution of 'a' for 'x' and 'b' for 'y' satisfies the equation, then the point (a, b) is a solution to the equation in coordinate terms.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can describe the position of any point using coordinates. This plane is defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. The point where these axes intersect is called the origin, labeled as \((0,0)\). Each point on the plane is represented by an ordered pair \((x, y)\), indicating its position relative to the origin.


Understanding the coordinate plane is foundational in coordinate geometry. It allows us to visualize mathematical equations by plotting their solutions as points or lines. Directions, distances, and areas can be represented and calculated using the geometry of the coordinate plane.

• The x-axis typically represents horizontal movement while the y-axis indicates vertical movement.
• Points above the x-axis have positive y-values, and those below have negative y-values.
• Points to the right of the y-axis have positive x-values, and those to the left have negative x-values.

Solution of an Equation

In mathematics, a solution to an equation is a set of values that make the equation true. When we talk about equations in two variables, such as \(x\) and \(y\), we often find pairs of \((x, y)\) values that satisfy the equation.

For example, if the equation is \(y = 2x + 3\), any pair of values for \(x\) and \(y\) that make both sides of the equation equal is considered a solution.

• To find a solution, substitute specific values into the equation and see if the equation holds true.
• If substituting \(a\) for \(x\) and \(b\) for \(y\) satisfies the equation, then the point \((a, b)\) is a solution.

The solution can be a point on the coordinate plane, illustrating that it lies on the graph of the equation.

Graph of an Equation

The graph of an equation in a coordinate plane is a visual representation of all its solutions. Every point on that graph can be a solution to the equation. For equations involving two variables \(x\) and \(y\), the graph might be a line, a curve, or another type of shape, depending on the form of the equation.

To graph an equation, follow these steps:

• Convert the equation into a form that allows you to express \(y\) directly in terms of \(x\) or vice versa.
• Choose values for \(x\) and solve for \(y\) to find points that satisfy the equation.
• Plot these points on the coordinate plane.
• Connect the points if the equation represents a continuous function.

The graph helps us understand the relationship between variables in an equation and easily find solutions just by looking at the curve or line.