Mathematics Review: Rational Equations, Cartesian Plane, Sequences, Distance and Midpoint

Solving Simple Rational Algebraic Equations Using Cross-Multiplication

• Definition of a Rational Algebraic Equation:

  • A rational algebraic equation is an equation containing one or more rational expressions, which are fractions in which both the numerator and the denominator are polynomials.

• The Cross-Multiplication Property:

  • For any proportion of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0$ and $d \neq 0$, cross-multiplication states that:     $a \cdot d = b \cdot c$
  • Cross-multiplication eliminates fractional denominators by multiplying each numerator by the denominator of the opposing side.

• Step-by-Step Procedure for Solving via Cross-Multiplication:

  • Step 1 (Identify Restrictions): Set each denominator equal to zero ($b = 0$ and $d = 0$) to determine values of the variable that cause the expression to be undefined.
  • Step 2 (Apply Cross-Multiplication): Multiply the numerator of the left side by the denominator of the right side, and set it equal to the product of the left denominator and right numerator ($a \cdot d = b \cdot c$).
  • Step 3 (Expand and Simplify): Use the distributive property to remove parentheses and combine like terms on both sides of the equation.
  • Step 4 (Solve the Algebraic Equation): Rearrange terms to isolate the variable using standard algebraic operations (for linear equations) or factoring / quadratic formula (for quadratic equations).
  • Step 5 (Check for Extraneous Solutions): Verify that the candidate solution does not make any original denominator equal to zero. If a candidate solution makes a denominator zero, it must be rejected as an extraneous root.

• Worked Examples:

  • Example 1 (Linear Outcome): Solve $\frac{x + 2}{3} = \frac{2x - 1}{5}$.

  • Step 1: Denominators are constants ($3$ and $5$), so there are no variable restrictions.

  • Step 2: Cross-multiply:       $5(x + 2) = 3(2x - 1)$

  • Step 3: Expand both sides:       $5x + 10 = 6x - 3$

  • Step 4: Isolate $x$:       $10 + 3 = 6x - 5x$$x = 13$

  • Step 5: Check solution: $\frac{13 + 2}{3} = \frac{15}{3} = 5$ and $\frac{2(13) - 1}{5} = \frac{25}{5} = 5$. The solution $x = 13$ is valid.

  • Example 2 (Quadratic Outcome): Solve $\frac{3}{x - 1} = \frac{x + 1}{4}$.

  • Step 1: Restriction is $x - 1 \neq 0 \implies x \neq 1$

  • Step 2: Cross-multiply:       $3 \times 4 = (x - 1)(x + 1)$

  • Step 3: Expand:       $12 = x^2 - 1$

  • Step 4: Solve for $x$:       $x^2 = 13$$x = \sqrt{13} \quad \text{or} \quad x = -\sqrt{13}$

  • Step 5: Both values satisfy $x \neq 1$, so both are valid solutions.

  • Example 3 (Extraneous Root Demonstration): Solve $\frac{x}{x - 2} = \frac{2}{x - 2}$.

  • Step 1: Restriction is $x - 2 \neq 0 \implies x \neq 2$

  • Step 2: Cross-multiply:       $x(x - 2) = 2(x - 2)$

  • Step 3: Expand and rearrange:       $x^2 - 2x = 2x - 4$$x^2 - 4x + 4 = 0$

  • Step 4: Factor:       $(x - 2)^2 = 0 \implies x = 2$

  • Step 5: Check restriction: $x = 2$ makes the denominator $x - 2 = 0$, making the expression undefined. Therefore, $x = 2$ is an extraneous solution, and the equation has no solution.

Illustration and Description of the Cartesian Coordinate Plane

• Definition and Historical Background:

  • The Cartesian coordinate plane is a two-dimensional grid system formed by two perpendicular real number lines intersecting at a right angle ($90^\circ$).
  • It was developed by French mathematician René Descartes to merge algebra and Euclidean geometry.

• Fundamental Components of the Cartesian Plane:

  • Horizontal Axis ($x$-axis / Abscissa Axis):
  • A horizontal number line where values increase from left to right.
  • Positive numbers lie to the right of the center; negative numbers lie to the left.
  • Vertical Axis ($y$-axis / Ordinate Axis):
  • A vertical number line where values increase from bottom to top.
  • Positive numbers lie above the center; negative numbers lie below.
  • Origin:
  • The point of intersection of the $x$-axis and $y$-axis.
  • Denoted by the capital letter $O$ with coordinates $(0, 0)$.

• Division into Quadrants:

  • The intersecting axes divide the Cartesian plane into four infinite regions called quadrants, numbered counterclockwise using Roman numerals:
  • Quadrant I ($QI$): Located in the upper-right corner. Both coordinates are positive ($x > 0$, $y > 0$), represented as $(+, +)$.
  • Quadrant II ($QII$): Located in the upper-left corner. The $x$-coordinate is negative and the $y$-coordinate is positive ($x < 0$, $y > 0$), represented as $(-, +)$.
  • Quadrant III ($QIII$): Located in the lower-left corner. Both coordinates are negative ($x < 0$, $y < 0$), represented as $(-, -)$.
  • Quadrant IV ($QIV$): Located in the lower-right corner. The $x$-coordinate is positive and the $y$-coordinate is negative ($x > 0$, $y < 0$), represented as $(+, -)$.

• Axis Boundaries:

  • Points that lie directly on the axes do not belong to any quadrant:
  • Points on the $x$-axis have a $y$-coordinate of zero: $(x, 0)$.
  • Points on the $y$-axis have an $x$-coordinate of zero: $(0, y)$.

Plotting Points and Determining Coordinates on the Cartesian Plane

• Concept of Ordered Pairs:

  • Every point on the Cartesian plane is uniquely identified by an ordered pair of real numbers expressed in the form $(x, y)$.
  • Abscissa ($x$): The first number in the ordered pair, representing the horizontal displacement and distance from the $y$-axis.
  • Ordinate ($y$): The second number in the ordered pair, representing the vertical displacement and distance from the $x$-axis.

• Step-by-Step Guide to Plotting Points $(x, y)$:

  • Step 1: Start at the origin $O(0, 0)$.
  • Step 2: Move horizontally along the $x$-axis by $|x|$ units:
  • Move to the right if $x > 0$.
  • Move to the left if $x < 0$.
  • Remain at zero if $x = 0$.
  • Step 3: From that horizontal position, move vertically by $|y|$ units:
  • Move upward if $y > 0$.
  • Move downward if $y < 0$.
  • Remain on the axis if $y = 0$.
  • Step 4: Draw a clear dot at the final location and label it with a capital letter and its coordinates, such as $P(x, y)$.

• Step-by-Step Guide to Determining Coordinates of a Given Point:

  • Step 1: Locate the given point on the plane.
  • Step 2: Trace a straight vertical line from the point to the $x$-axis. Read the value at the intersection to determine the $x$-coordinate.
  • Step 3: Trace a straight horizontal line from the point to the $y$-axis. Read the value at the intersection to determine the $y$-coordinate.
  • Step 4: Write the two values as an ordered pair $(x, y)$.

• Categorization of Coordinate Examples:

  • Point $A(4, 5)$: Lies in Quadrant I ($x > 0$, $y > 0$).
  • Point $B(-3, 6)$: Lies in Quadrant II ($x < 0$, $y > 0$).
  • Point $C(-2, -7)$: Lies in Quadrant III ($x < 0$, $y < 0$).
  • Point $D(5, -4)$: Lies in Quadrant IV ($x > 0$, $y < 0$).
  • Point $E(0, -3)$: Lies on the negative $y$-axis.
  • Point $F(6, 0)$: Lies on the positive $x$-axis.

Formulating Sequence Rules and Predicting Subsequent Terms

• Fundamentals of Mathematical Sequences:

  • A sequence is an ordered set of numbers structured according to a specific rule or pattern.
  • Each number in the sequence is termed a term and is denoted as $a_n$, where $n \in \{1, 2, 3, \dots\}$ represents the term's position index.

• Arithmetic Sequences (Linear Patterns):

  • Definition: A sequence where the difference between consecutive terms is constant.
  • Common Difference Formula:     $d = a_n - a_{n-1}$
  • General Explicit Formula:     $a_n = a_1 + (n - 1)d$
  • Example Analysis:
  • Sequence: $3, 7, 11, 15, 19, \dots$
  • Calculate common difference: $7 - 3 = 4$, $11 - 7 = 4$, $15 - 11 = 4$. Thus, $d = 4$
  • Formulate rule: $a_n = 3 + (n - 1)4 = 4n - 1$
  • Predict next term ($a_6$): $a_6 = 19 + 4 = 23$ or $a_6 = 4(6) - 1 = 23$

• Geometric Sequences (Exponential Patterns):

  • Definition: A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Common Ratio Formula:     $r = \frac{a_n}{a_{n-1}}$
  • General Explicit Formula:     $a_n = a_1 \cdot r^{n-1}$
  • Example Analysis:
  • Sequence: $2, 6, 18, 54, \dots$
  • Calculate common ratio: $\frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$. Thus, $r = 3$
  • Formulate rule: $a_n = 2 \cdot 3^{n-1}$
  • Predict next term ($a_5$): $a_5 = 54 \times 3 = 162$ or $a_5 = 2 \cdot 3^{5-1} = 2 \cdot 81 = 162$

• Quadratic Sequences (Second-Difference Patterns):

  • Definition: A sequence where the first differences between consecutive terms change, but the second differences (the differences between the first differences) remain constant.
  • General Explicit Formula:     $a_n = an^2 + bn + c$
  • Example Analysis:
  • Sequence: $2, 5, 10, 17, 26, \dots$
  • Compute first differences: $5 - 2 = 3$, $10 - 5 = 5$, $17 - 10 = 7$, $26 - 17 = 9$
  • Compute second differences: $5 - 3 = 2$, $7 - 5 = 2$, $9 - 7 = 2$
  • Next first difference: $9 + 2 = 11$
  • Predict next term ($a_6$): $26 + 11 = 37$
  • Formulate rule: $a_n = n^2 + 1$

• Special Sequences and Exponential/Power Patterns:

  • Square Numbers: $1, 4, 9, 16, 25, \dots$ with rule $a_n = n^2$. Next term $a_6 = 36$
  • Cube Numbers: $1, 8, 27, 64, 125, \dots$ with rule $a_n = n^3$. Next term $a_6 = 216$
  • Alternating Sign Sequences: $1, -2, 4, -8, 16, \dots$ with rule $a_n = (-1)^{n-1} \cdot 2^{n-1}$. Next term $a_6 = -32$

• General Strategy for Identifying Patterns:

  • Step 1: Calculate the first differences between successive terms ($a_n - a_{n-1}$).
  • Step 2: If first differences are constant, apply the arithmetic formula.
  • Step 3: If first differences are not constant, check the ratio of consecutive terms ($\frac{a_n}{a_{n-1}}$). If constant, apply the geometric formula.
  • Step 4: If ratios are not constant, calculate second differences. If second differences are constant, apply quadratic analysis.
  • Step 5: Test the deduced rule against all given terms in the sequence before writing out the subsequent term.

Distance and Midpoint Formulas on the Cartesian Plane

• The Distance Formula:

  • Definition: Calculates the exact Euclidean length of the line segment connecting two arbitrary points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the Cartesian coordinate plane.
  • Derivation: Based on the Pythagorean Theorem ($c^2 = a^2 + b^2$) applied to a right triangle formed by horizontal legs of length $|x_2 - x_1|$ and vertical legs of length $|y_2 - y_1|$.
  • Formula:     $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  • Simplified Cases for Horizontal and Vertical Segments:
  • Horizontal Line Segment ($y_1 = y_2$):       $d = |x_2 - x_1$
  • Vertical Line Segment ($x_1 = x_2$):       $d = |y_2 - y_1$
  • Worked Example: Calculate the distance between $A(-3, 4)$ and $B(5, -2)$.
  • Assign coordinates: $x_1 = -3$, $y_1 = 4$, $x_2 = 5$, $y_2 = -2$
  • Substitute into the formula:       $d = \sqrt{(5 - (-3))^2 + (-2 - 4)^2}$$d = \sqrt{(5 + 3)^2 + (-6)^2}$$d = \sqrt{8^2 + (-6)^2}$$d = \sqrt{64 + 36}$$d = \sqrt{100} = 10$

• The Midpoint Formula:

  • Definition: Determines the exact coordinates of the middle point $M$ that bisects a line segment joining endpoints $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$.
  • Conceptual Foundation: The midpoint coordinates represent the arithmetic average of the $x$-coordinates and the arithmetic average of the $y$-coordinates of the endpoints.
  • Formula:     $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
  • Worked Example 1 (Finding Midpoint from Endpoints): Find the midpoint $M$ of the line segment connecting $A(-3, 4)$ and $B(5, -2)$.
  • Assign coordinates: $x_1 = -3$, $y_1 = 4$, $x_2 = 5$, $y_2 = -2$
  • Apply the formula:       $M = \left(\frac{-3 + 5}{2}, \frac{4 + (-2)}{2}\right)$$M = \left(\frac{2}{2}, \frac{2}{2}\right) = (1, 1)$
  • Worked Example 2 (Finding an Endpoint given One Endpoint and Midpoint):
  • Problem: Point $A(2, -5)$ is one endpoint of segment $AB$. The midpoint of segment $AB$ is $M(6, 1)$. Find the coordinates of endpoint $B(x_2, y_2)$.
  • Set up component equations:       $\frac{2 + x_2}{2} = 6 \implies 2 + x_2 = 12 \implies x_2 = 10$$\frac{-5 + y_2}{2} = 1 \implies -5 + y_2 = 2 \implies y_2 = 7$
  • Solution: Endpoint $B$ has coordinates $(10, 7)$.

Your notes are available right here in our chat context! They cover five main topics in mathematics: solving simple rational algebraic equations using cross-multiplication, describing and plotting points on the Cartesian coordinate plane, formulating sequence rules, and using the distance and midpoint formulas.