Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each point lies on the graph of the equation. \(x^{2}+y^{2}=20\) (a) (3,-2) (b) (-4,2)

Step-by-Step Solution

Answer

Point (a) (3,-2) does not lie on the graph of the equation whereas point (b) (-4,2) does.

Step 1: Substitute the coordinates of the first point

We plug the x and y coordinates from point (a) (3,-2) into the equation. This gives us \(3^{2} + (-2)^{2} = 20\). After simplification, this simplifies to \(9 + 4 = 13\). Since 13 does not equal 20, we can conclude that point (a) (3,-2) does not lie on the graph of the equation.

Step 2: Substitute the coordinates of the second point

Next, we substitute the x and y coordinates from point (b) (-4,2) into the equation to get \((-4)^{2} + 2^{2} = 20\). After simplifying this, we get \(16 + 4 = 20\). Since both sides are equal, we conclude that point (b) (-4,2) does lie on the graph of the equation.

Key Concepts

Coordinate Geometry

Coordinate geometry, often called analytic geometry, blends algebra and geometry by using symbolic representations and numerical values to solve geometric problems. It describes shapes like lines, triangles, and circles on a plane, using a coordinate system, typically the Cartesian coordinate system which is composed of two axes: the x-axis (horizontal) and the y-axis (vertical).

In the Cartesian plane, a point is identified by its coordinates \(x, y\). The x-coordinate represents its horizontal position, while the y-coordinate represents its vertical position. In coordinate geometry, we can describe curves and shapes with equations. The equation \(x^2 + y^2 = 20\) describes a circle centered at the origin \(0, 0\) with a radius of \sqrt{20}\.

Points are tested to see if they lie on a particular shape by substituting their coordinates into an equation. If the equation holds true, the point lies on the graph of the equation. Otherwise, it doesn't.

Substitution Method

The substitution method allows us to determine if a point is part of a specific geometric figure represented by an equation. We do this by injecting the point's x and y values into the given equation and checking the equation's validity.

For instance, with the equation \(x^2 + y^2 = 20\), to verify if the point \(3, -2\) lies on the graph, substitute 3 for x and -2 for y. Calculating, we get \(3^2 + (-2)^2 = 9 + 4 = 13\). Since 13 does not equal 20, \(3, -2\) is not on the graph. This simple plug-and-check process makes substitution easy to grasp and use.

Similarly, for the point \(-4, 2\), substituting gives us \((-4)^2 + 2^2 = 16 + 4 = 20\). Here, 20 equals 20, confirming that the point lies on the graph. Substitution provides a straightforward path to verify the inclusion of points in any graph of an equation.

Equation Verification

Equation verification is the process of checking whether a mathematical statement or solution is true or valid. By evaluating whether substituted points satisfy the equation's conditions, we confirm if these points correctly represent the geometry described.

In our example, we considered the equation \(x^2 + y^2 = 20\). Equation verification required checking given points, \(3, -2\) and \(-4, 2\), against this circle's equation. \(3, -2\) failed to satisfy the condition as it resulted in 13, not 20. Thus, it doesn't lie on the circle.

Conversely, point \(-4, 2\) resulted in a valid equation check, verifying that it does lie on the circle because the calculations yielded 20, matching both sides of the equation. Accurate substitution followed by verification ensures solutions are correct, playing a crucial role in coordinate geometry and beyond.