Problem 49 - Precalculus Mathematics for Calculus

Question

Determine whether the given points are on the graph of the equation. $$x-2 y-1=0 ; \quad(0,0),(1,0),(-1,-1)$$

Step-by-Step Solution

Answer

(1,0) and (-1,-1) are on the graph, but (0,0) is not.

Step 1: Understand the Equation

We are given the equation \(x - 2y - 1 = 0\). This is a linear equation in two variables, \(x\) and \(y\). To determine if a point \((a, b)\) lies on this graph, we should substitute \(x = a\) and \(y = b\) into the equation and check if the equation holds true.

Step 2: Test Point (0,0)

Substitute \(x = 0\) and \(y = 0\) into the equation. This gives us \(0 - 2(0) - 1 = 0 - 1 = -1\). Since \(-1 eq 0\), the point \((0,0)\) is not on the graph.

Step 3: Test Point (1,0)

Substitute \(x = 1\) and \(y = 0\) into the equation. This gives us \(1 - 2(0) - 1 = 1 - 1 = 0\). Since \(0 = 0\), the point \((1,0)\) is on the graph.

Step 4: Test Point (-1,-1)

Substitute \(x = -1\) and \(y = -1\) into the equation. This gives us \(-1 - 2(-1) - 1 = -1 + 2 - 1 = 0\). Since \(0 = 0\), the point \((-1,-1)\) is on the graph.

Key Concepts

Graph of an Equation

To understand if certain points lie on the graph of a linear equation, imagine a straight line on a 2D plane. This line represents all possible solutions for the variables in the equation. For example, with the equation \(x - 2y - 1 = 0\), any combination of \(x\) and \(y\) that satisfies this equation will be on this line.

The process of plotting a graph for an equation helps us visually interpret and validate solutions. Each point on this line is a solution, and the equation defines the relationship between the \(x\) and \(y\) coordinates. For any given \(x\), you can solve for \(y\) to find a corresponding point on the graph. Conversely, for any \(y\), you can solve for \(x\).

• A point like \((x, y)\) is considered to be on the graph if substituting \(x\) and \(y\) into the equation makes it true (both sides equal).
• If substitution reveals the equation is false (both sides unequal), then that point is not on the graph.

Substitution Method

The substitution method is a valuable tool for verifying if a point lies on the graph of an equation. Here's how it works:

Consider the linear equation \(x - 2y - 1 = 0\). Given a point, such as \((1, 0)\), substitute \(x = 1\) and \(y = 0\) into the equation. Perform the arithmetic to see if both sides match.

• For \((1, 0)\): Substitute into the equation to get \(1 - 2(0) - 1 = 0\). Simplify this to see everything balances out to zero, hence the point is on the graph.
• If when you substitute a point, the left-hand side equals a number other than zero, that point doesn't belong on the graph.

Using this method is straightforward for linear equations. Replace variables with their numerical values from the point and solve. This will quickly show if you're dealing with a point on the line.

For more complex equations or systems with multiple equations, the substitution method can also be used to simplify and solve the equations by eliminating variables.

Solution Verification

Verifying a solution ensures accuracy and reinforces understanding. In this context, solution verification means confirming whether a point is indeed part of the graph of the equation.

After performing the substitution method, it's critical to review your arithmetic carefully. Mistakes in substitution or calculation can lead to incorrect conclusions about whether a point lies on a graph.

• Always re-evaluate the solution by casting a second look at the calculated steps.
• Use mental math or a calculator to double-check your findings.
• Sometimes graphing the equation on paper or with software can offer a visual check to see if the points line up rest on the graph.

This process not only checks the math but also deepens the understanding of the relationship between algebraic expressions and their graphical representation.