Problem 4 - Precalculus: Mathematics for Calculus

Question

If the point 12, 32 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 \(2 y=x+1 ?\) Complete the table, and sketch a graph.

Step-by-Step Solution

Answer

The point (12, 32) is not on the graph of the equation since 64 ≠ 13.

Step 1: Identify the Values of x and y

The given point is (12, 32). This means that the value of x is 12 and the value of y is 32. In the equation, replace x with 12, and y with 32.

Step 2: Substitute Values into the Equation

Use the equation given: \(2y = x + 1\). Substitute x = 12 and y = 32 into this equation. The equation becomes: \[2(32) = 12 + 1\].

Step 3: Solve the Equation

Perform the calculations from the substitution step: \[64 = 13\]. This equation is false, meaning the left-hand side does not equal the right-hand side.

Step 4: Determine if the Point is on the Graph

Since the equation \(64 = 13\) is not true, the point (12, 32) is not on the graph of the equation \(2y = x + 1\).

Step 5: Complete the Table and Sketch the Graph

To create a table, pick a few values of x, compute corresponding y values with the equation \(y = \frac{x+1}{2}\). For example, if x = 0, y = 0.5; if x = 2, y = 1.5; if x = 4, y = 2.5. Plot these points on a graph and draw a straight line through them to represent the equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in coordinate geometry. A linear equation in two variables can be written in various forms, such as standard form, slope-intercept form, or a rearranged form. The simplest way to graph such an equation is by finding two points that satisfy it, then plotting these points on the coordinate plane.
Once you have two points, you can draw a straight line through them. This line represents all the solutions that make the equation true. The visual representation allows for easy interpretation of linear relationships between variables.
To practice, pick random values for x, compute their corresponding y values using the equation, and plot the results. As was done in the example, choose values like x = 0, x = 2, and x = 4 to obtain points like (0, 0.5), (2, 1.5), and (4, 2.5). Then draw the line through these points, which will help illustrate the equation.

Substitution Method

The substitution method is used to check if a specific point lies on the graph of an equation. When given a point with coordinates (x, y), substitute these values into the equation and simplify to see if the equation holds true.

This method helps confirm whether a certain combination of x and y satisfies the equation. If after substitution, the equation is balanced, then the point is indeed on the line represented by the equation. This was demonstrated in the original exercise where the point (12, 32) was substituted into the equation \(2y = x + 1\).

• Replace x with 12 and y with 32, resulting in: \(2 \cdot 32 = 12 + 1\).
• Solve to find that \(64 eq 13\), confirming that the point is not on the graph.

Using substitution is a straightforward method to check the validity of solutions in linear equations.

Checking Solutions in Equations

Checking solutions in equations involves verifying if the substituted values satisfy the equation. This is a key skill in algebra and helps in understanding whether or not given points are indeed part of the graphical solutions of the equation.
By substituting and solving the equation, one can determine the validity of a point. For the example equation \(2y = x + 1\), a solution is only correct if both sides of the equation equal after substitution.

• If substituting x = 12 and y = 32 gives an incorrect equation like \(64 = 13\), then the point is not part of the solution.
• Repeat this process for any point to verify its place on the graph.

This approach also builds confidence in exploring equations, providing a systematic way to confirm or find errors in given solutions.