Problem 4 - Precalculus Mathematics for Calculus

Question

If the point (2, 3) is on the graph of an equation in x and y, then the equation is satisfied when we replace x by _______ and y by _______ Is the point 12, 32 on the graph of the equation \(2 y=x+1 ?\)

Step-by-Step Solution

Answer

x is 2, y is 3. The point (12, 32) is not on the graph.

Step 1: Understanding the Equation

First, we must understand what the equation is asking us. The equation given is \(2y = x + 1\). Any point on this equation must satisfy this relationship between \(x\) and \(y\).

Step 2: Identifying the Replacement Values

From the problem, when we are given the point \((2, 3)\), \(x\) is replaced by 2 and \(y\) is replaced by 3. Thus, \(x = 2\) and \(y = 3\).

Step 3: Checking if the Point Satisfies the Equation

Now, we substitute \(x = 12\) and \(y = 32\) into the equation \(2y = x + 1\). This results in \(2(32) = 12 + 1\).

Step 4: Calculating Both Sides of the Equation

Calculate both sides: \(2(32) = 64\) and \(12 + 1 = 13\).

Step 5: Comparing the Results

Compare the results: since \(64 eq 13\), the point \((12, 32)\) does not satisfy the equation \(2y = x + 1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that utilizes a coordinate system to investigate geometric properties and relationships. In this system, a point is defined by a pair of numerical values, often referred to as \(x\) and \(y\) coordinates. These coordinates describe a specific location on a two-dimensional plane.
For instance, the point \( (2, 3) \) is located by moving 2 units along the x-axis and 3 units along the y-axis. This visualization helps in realizing how every point on a graph satisfies a particular mathematical condition represented by an equation.
Such equations typically relate x and y coordinates in a way that creates a recognizable curve or line when plotted. Understanding this relationship is key to solving problems in coordinate geometry and algebra.

Substitution Method

The substitution method is a powerful technique to solve systems of equations algebraically. In this method, you solve one of the equations for one of the variables and then substitute this expression into the other equation. This process simplifies the equation into a single variable, making it easier to solve.
In the context of a single equation, like \(2y = x + 1\), you substitute specific values of x and y into the equation to test if a point lies on the line. For example, to check if the point \( (12, 32) \) satisfies the equation, substitute \( x = 12 \) and \( y = 32 \):

\[ 2(32) = 12 + 1 \]

After substitution, if both sides of the equation are equal, the point is on the line. If not, as is the case here, the point is not part of the graph derived from the equation.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy a linear equation, such as \(2y = x + 1\). A linear equation represents a straight line in the plane, and any point on this line fulfills the equation.
To graph a linear equation, you typically start by finding points that satisfy the equation. This is done by selecting values for one variable (usually x), calculating the corresponding y values using the equation, and then plotting these points. Connecting the dots forms a straight line.
Visualizing equations this way helps identify the relationship between the variables and interpret the solution graphically. For example, graphing \(2y = x + 1\) would reveal a line where every point (\

(2y - 1)/2, y\] \\( satisfies the equation. This visual approach enhances understanding of the structure and behavior of linear relationships in mathematics.