Problem 1 - Algebra and Trigonometry

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 \((2,3)\) on the graph of the equation \(2 y=x+1 ?\)

Step-by-Step Solution

Answer

Replace \(x\) by 2 and \(y\) by 3; the point is not on the graph.

Step 1: Identify the Known Values

The point given is \((2,3)\). Here, the coordinates \((2,3)\) represent \(x = 2\) and \(y = 3\). That means wherever there is \(x\) in the equation, it will be replaced by \(2\), and wherever there is \(y\), it will be replaced by \(3\).

Step 2: Substitute the Values into the Equation

The equation provided is \(2y = x + 1\). Substitute \(x = 2\) and \(y = 3\) into this equation. So, we replace \(y\) with \(3\) and \(x\) with \(2\), resulting in: \(2(3) = 2 + 1\).

Step 3: Simplify and Verify the Equation

Calculate the left side of the equation: \(2 \times 3 = 6\). Now calculate the right side: \(2 + 1 = 3\). The equation \(6 = 3\) does not hold true, therefore, the point \((2,3)\) is not on the graph of the equation \(2y = x + 1\).

Key Concepts

Understanding Coordinates

Coordinates are a pair of numbers used to uniquely identify a point's position on a graph. In our exercise, the point is given as \((2,3)\). These numbers refer to values on the x-axis (2) and y-axis (3).

Whenever you are asked to verify if a point is on the graph of an equation, you use these coordinates to transform the equation into a statement that can be either true or false. It’s like a mathematical address, with \(x = 2\) and \(y = 3\) acting as your guiding lights for substitution into the equation.

• x-coordinate: This represents the horizontal position of the point.
• y-coordinate: This represents the vertical position of the point.

Substitution Method

The substitution method involves replacing variables with specific values. In this problem, you're asked to use the substitution method to determine if a point is part of an equation’s graph.

In the equation \(2y = x + 1\), the task is to substitute \(x = 2\) and \(y = 3\). This turns the original equation into \(2(3) = 2 + 1\). The main goal is simplification since it allows you to evaluate if both sides are equal.

• Identify the variables in the equation.
• Replace with known coordinates: substitute each variable with its given value.
• Simplify: perform calculations to verify the elements of the equation.

Verification of a Point

Verification of a point on a graph means confirming whether the substituted values satisfy the equation. After performing substitution in the exercise, the equation became \(6 = 3\).

Clearly, this calculation doesn't equal, showing that our point doesn’t lie on the graph of the equation \(2y = x + 1\). This task reveals a key skill in graphing: ensuring your point meets the equation’s conditions.

• Calculate both sides of the equation separately.
• Check if both values are equal post-calculation.
• If equal, the point is on the graph; if not, it isn't.

Verification is essential in confirming graphical integrity; it establishes a bridge between a mathematical equation and its graphical representation.