Problem 6 - Precalculus

Question

Determine whether the given point lies on the graph of the equation, as in Example \(1 .\) Note: You are not asked to draw the graph. $$(a-1, a+1) ; y=x+2$$

Step-by-Step Solution

Answer

Yes, the point lies on the line.

Step 1: Understand the Problem

We need to check if the point \((a - 1, a + 1)\) lies on the line described by the equation \(y = x + 2\). To do this, substitute \(x = a - 1\) and \(y = a + 1\) into the equation.

Step 2: Substitute the x-value

Substitute \(x = a - 1\) into the equation \(y = x + 2\). This gives us an expression for \(y\): \[ y = (a - 1) + 2. \]

Step 3: Simplify the y-expression

Simplify the expression \(y = (a - 1) + 2\) to find the value of \(y\). \[ y = a - 1 + 2 = a + 1. \]

Step 4: Substitute and Verify y-value

Now check if the simplified value \(y = a + 1\) matches the y-coordinate of the point \((a - 1, a + 1)\). Since both values are \(a + 1\), the point satisfies the equation.

Step 5: Conclusion

Since substituting \(x = a - 1\) into the equation yielded \(y = a + 1\), the point \((a - 1, a + 1)\) indeed lies on the line \(y = x + 2\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric shapes. It's a fascinating way to describe the spatial relationships between points, lines, and figures on a plane. In coordinate geometry, every point is defined by a pair of numbers: its coordinates. These coordinates tell you exactly where a point lies on the two-dimensional plane.

In a simple coordinate plane, the horizontal and vertical axes are called the x-axis and y-axis, respectively. The point where they intersect is the origin, labeled as \(0, 0\). The position of any point is told by its coordinates, specifically \(x, y\). Lines in this system can be described with equations, where y often depends on changes in x.

• The equation of a line is often written in the form \(y = mx + b\), where \m\ is the slope and \b\ is the y-intercept.
• This slope-intercept form helps us to understand how the line moves across the plane and where it crosses the y-axis.

Coordinate geometry is powerful because it links algebra and geometry, letting you calculate distances, midpoints, and areas of figures using formulas. This makes solving problems involving shapes and their properties much simpler and more systematic.

Substitution Method

The substitution method is a straightforward yet effective technique for solving equations, particularly those that involve checking points on a line. Here, we apply it to determine whether a certain point lies on a given line by substituting the point's coordinates into the line's equation.

First, identify the given point's coordinates. For example, with the point \(a - 1, a + 1\), \a - 1\ is your x-value and \a + 1\ is your y-value. Your task is to substitute these values into the line's equation, which is \(y = x + 2\) in this exercise.

• Substitute the x-coordinate into the equation: Replace \(x\) with \(a - 1\). This becomes \(y = (a - 1) + 2\).
• Simplify the result: In this example, simplifying gives \(y = a + 1\).

Once you have the simplified result, compare it to the y-coordinate given by the point. If these match, then the point lies on the line—demonstrating that it satisfies the line's equation when using substitution.

Verification of Points

Verification of points on a geometric line involves confirming that the coordinates of a given point satisfy the equation of that line. This process helps ensure that any point you claim is on the line, indeed, fulfills the line's properties.

The process starts by using the substitution method, as explained earlier. After substituting the point's x-coordinate into the equation and simplifying, compare the resulting y-value to the y-coordinate of the point in question.

• If both y-values match, the point lies on the line.
• If they don't match, the point is not on the line.

Verification is crucial since it prevents errors when plotting graphs or solving geometric problems, ensuring accuracy and confirming the points are authentically part of the line. This confirmation step is simple yet essential, providing certainty that the math holds up under scrutiny.