Problem 27 - Elementary and Intermediate Algebra

Question

Consecutive Integer Problems Plotting Points. The \(x\) -coordinate and \(y\) -coordinate of a point in quadrant I are consecutive odd integers whose product is \(143 .\) The \(x\) -coordinate is less than the \(y\) -coordinate. Find the coordinates of the point.

Step-by-Step Solution

Answer

The coordinates of the point are \((11, 13)\).

Step 1: Understand the Problem

We are given two consecutive odd integers in the first quadrant of a coordinate plane where one number is the \(x\)-coordinate and the other is the \(y\)-coordinate. The product of these two integers is \(143\), and the \(x\)-coordinate is less than the \(y\)-coordinate.

Step 2: Express the Odd Integers

Let the first odd integer be \(x\). Since the problem states the integers are consecutive and odd, we can express the \(y\)-coordinate as \(x + 2\) because that's the next odd number after \(x\).

Step 3: Set Up the Equation

According to the problem, the product of the two coordinates \(x\) and \(y\) (which is \(x + 2\)) is \(143\). Thus, the equation is: \[ x(x + 2) = 143 \]

Step 4: Expand the Equation

Expand the left-hand side of the equation: \[ x^2 + 2x = 143 \]

Step 5: Rearrange into a Quadratic Equation

Subtract \(143\) from both sides to set the equation to zero: \[ x^2 + 2x - 143 = 0 \]

Step 6: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation, \(a = 1\), \(b = 2\), and \(c = -143\), giving us: \[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-143)}}{2 \cdot 1} \]

Step 7: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac = 4 + 572 = 576\).

Step 8: Simplify the Quadratic Formula

Substitute back the value \(576\) to solve for \(x\): \[ x = \frac{-2 \pm \sqrt{576}}{2} \]. Simplifying, \(\sqrt{576} = 24\), resulting in: \[ x = \frac{-2 \pm 24}{2} \].

Step 9: Find Possible Values of \(x\)

Calculating, we get: \( x = \frac{22}{2} = 11 \) or \( x = \frac{-26}{2} = -13 \). Since we need positive integers in Quadrant I, \(x = 11\) is our solution.

Step 10: Determine the \(y\)-coordinate

Since \(x = 11\), then \(y = x + 2 = 13\). Hence, the coordinates of the point are \((11, 13)\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions where the highest exponent of the variable is squared. These are typically in the standard form:

• \[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The highest power of \( x \) is 2, which gives it its name—the quadratic equation. Solving these equations often involves finding the values of \( x \) which make the equation true.

There are different methods to solve quadratic equations:

• Factoring
• Completing the Square
• Using the Quadratic Formula

The quadratic formula is quite powerful and can solve any quadratic equation. It is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Using this formula, we can find the roots or solutions of the quadratic equation, which can be either real or complex numbers.

In our exercise, through each step we explored the quadratic equation by inserting the values of \( a \), \( b \), and \( c \), revealing the two solutions. Understanding the roots helped us solve the coordinates correctly as only the positive value for \( x \) was relevant to our first quadrant point.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane where each point is determined by a pair of numerical coordinates. These are defined as the distances to two perpendicular axes:

• the "x"-axis horizontally
• the "y"-axis vertically

These axes divide the plane into four quadrants allowing for precise location of points using pairs like \((x, y)\).

The use of coordinate planes is essential in various mathematical concepts, including graphing equations and data analysis. Each point on this plane represents a unique position formed by associating a particular "x" value with a "y" value.

In consecutive integer problems, such as the exercise above, coordinate planes help visualize solutions or determine where certain conditions, like products of coordinates being equal to a specific value, hold true. By plotting on the coordinate plane, we observe how these conditions manifest, thus aiding in a comprehensive understanding of the solution.

First Quadrant

The first quadrant of the coordinate plane is the top-right section, where both the "x" and "y" coordinates are positive. It's one of the four sections created when the "x" and "y" axes intersect at the origin \(0,0\). Here's a breakdown of its characteristics:

• Both coordinates \(x\) and \(y\) are positive.
• Typical area of focus for many basic real-world problems because they simplify the workability of the solutions.
• Arithmetic operations and solutions usually result in tangible "real-world" outputs suitable for initial learning contexts.

Understanding the first quadrant is essential for students beginning to explore graphing points and solving equations through visual representation. In our consecutive integer problem, the solution focused on this quadrant because both \(x\) and \(y\) were required to be positive integers only. This constraint simplifies the problem and narrows down potential solutions, as seen by the valid solution of \( (11, 13) \). In essence, the said quadrant provides a meaningful context that ensures our solutions make real-world sense within the problem's restrictions.