Algebraic Equations & Geometry: Solving Equations and Studying Shapes, Study notes of Mathematics

Download Algebraic Equations & Geometry: Solving Equations and Studying Shapes and more Study notes Mathematics in PDF only on Docsity! Algebraic Equations & Geometry Notes Examples 1. Introduction Introduction to Algebraic Equations and Geometry Algebraic equations and geometry are fundamental branches of mathematics that deal with solving equations and studying shapes, sizes, and properties of objects in space. They have widespread applications in various fields such as engineering, physics, economics, and computer science. Understanding algebraic equations and geometry is crucial for problem-solving and critical thinking skills development. Importance and Applications in Mathematics and Real Life Algebraic equations help in modeling and solving real-world problems involving unknown quantities. Geometry is essential for understanding spatial relationships, construction, and measurement. Both algebraic equations and geometry play vital roles in technology, architecture, art, and everyday life. 2: Algebraic Equations Section 1: Linear Equations Definition and Examples A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power. Example: \(3x + 2 = 8\) is a linear equation. Solving Methods Substitution Substitution Method: - Step 1: Solve one of the equations for one variable in terms of the other variable. - Step 2: Substitute the expression found in step 1 into the other equation. - Step 3: Solve the resulting equation for the remaining variable. - Example: - Equations: \(2x + 3y = 7\) and \(x - y = 2\) - Solve the second equation for \(x\): \(x = y + 2\) - Substitute \(x\) into the first equation: \(2(y + 2) + 3y = 7\) - Solve for \(y\) and then find \(x\). Elimination Elimination Method: - Step 1: Multiply one or both equations by appropriate constants to eliminate one variable when added or subtracted. - Step 2: Add or subtract the equations to eliminate one variable. - Step 3: Solve the resulting equation for the remaining variable. - Example: - Equations: \(2x + 3y = 7\) and \(4x - 2y = 6\) - Multiply the second equation by 3: \(12x - 6y = 18\) - Subtract the first equation from the modified second equation: \(10x = 11\) - Solve for \(x\) and then find \(y\). Graphing Graphing Method: - Step 1: Graph each equation on the coordinate plane. - Step 2: Identify the point of intersection, which represents the solution to the system. - Example: - Equations: \(2x + 3y = 7\) and \(x - y = 2\) - Graph both equations and find their intersection point. 3: Algebraic Equations (Continued) Section 2: Quadratic Equations Definition and Examples A quadratic equation is a second-degree polynomial equation in a single variable with the general form \(ax^2 + bx + c = 0\), where \(a \neq 0\). Example: \(x^2 - 4x + 4 = 0\) is a quadratic equation. Solving Method Factoring Factoring Method: - Step 1: Rewrite the equation in the form \(ax^2 + bx + c = 0\). - Step 2: Factor the quadratic expression on the left side of the equation. - Step 3: Set each factor equal to zero and solve for \(x\). - Example: - Equation: \(x^2 - 4x + 4 = 0\) - Factor the expression: \((x - 2)^2 = 0\) - Set each factor equal to zero: \(x - 2 = 0 \Rightarrow x = 2\) Quadratic Formula Quadratic Formula: - The quadratic formula is given by \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation. - Example: - Equation: \(x^2 - 4x + 4 = 0\) - Using the quadratic formula: \(x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4(1)(4)}}}}{{2 (1)}}\) - Simplify and solve for \(x\). Completing the Square Completing the Square Method: - Step 1: Rewrite the equation in the form \(ax^2 + bx + c = 0\), if necessary. - Step 2: Move the constant term to the other side of the equation. - Step 3: Add and subtract the square of half of the coefficient of the linear term. - Step 4: Factor the perfect square trinomial and solve for \(x\). - Example: - Equation: \(x^2 - 4x + 4 = 0\) - Complete the square: \(x^2 - 4x + 4 - 4 = 0\) - Factor the perfect square trinomial: \((x - 2)^2 = 0\) - Solve for \(x\). 4: Algebraic Equations (Continued) Section 3: Systems of Equations Definition and Examples A system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the set of values that satisfy all equations in the system Example: System of Equations: 1. \(2x + 3y = 10\) 2. \(x - 2y = 5\) Solving Methods: Substitution Substitution Method: - Step 1: Solve one of the equations for one variable in terms of the other variable. - Step 2: Substitute the expression found in step 1 into the other equation. - Step 3: Solve the resulting equation for the remaining variable. - Example: - Equations: \(2x + 3y = 10\) and \(x - 2y = 5\) - Solve the second equation for \(x\): \(x = 2y + 5\) - Substitute \(x\) into the first equation: \(2(2y + 5) + 3y = 10\) - Solve for \(y\) and then find \(x\). Elimination Elimination Method: - Step 1: Multiply one or both equations by appropriate constants to eliminate one variable when added or subtracted. - Step 2: Add or subtract the equations to eliminate one variable. - Step 3: Solve the resulting equation for the remaining variable. - Example: - Equations: \(2x + 3y = 10\) and \(4x - 6y = 20\) - Multiply the first equation by 2 and the second equation by 3: \(4x + 6y = 20\) and \(12x - 18y = 60\) - Add the equations to eliminate \(y\): \(16x = 80\) - Solve for \(x\) and then find \(y\). Graphing Graphing Method: - Step 1: Graph each equation on the coordinate plane. - Step 2: Identify the point of intersection, which represents the solution to the system. - Example: - Equations: \(2x + 3y = 10\) and \(x - 2y = 5\) - Graph both equations and find their intersection point. 5: Geometry Section 4: Basic Geometry Concepts Points, Lines, and Planes Points: A point is a precise location in space, represented by a dot. It has no size, length, or width. Lines: A line is a straight path that extends infinitely in both directions. It is defined by two points. Planes: A plane is a flat surface that extends infinitely in all directions. It is defined by three non-collinear points. Example: In the coordinate plane, point A(2,3) and point B(5,7) determine a line AB, while points A, B, and C determine a plane ABC. Angles and Lines: Classification and Properties Angles: An angle is formed by two rays with a common endpoint, called the vertex. Classification of Angles: 1. Acute Angle: An angle whose measure is less than 90 degrees. 2. Right Angle: An angle whose measure is exactly 90 degrees. 3. Obtuse Angle: An angle whose measure is greater than 90 degrees but less than 180 degrees. 4. Straight Angle: An angle whose measure is exactly 180 degrees. 5. Reflex Angle: An angle whose measure is greater than 180 degrees but less than 360 degrees. Properties of Angles: - Adjacent Angles: Two angles that share a common side and a common vertex, but no common interior points. - Vertical Angles: Two non-adjacent angles formed by intersecting lines. They are congruent. - Complementary Angles: Two angles whose measures add up to 90 degrees. - Supplementary Angles: Two angles whose measures add up to 180 degrees. Example: In triangle ABC, angle A and angle B are adjacent angles, while angle A and angle C are complementary angles if the measure of angle A is 30 degrees and the measure of angle C is 60 degrees. 6: Geometry (Continued) Section 5: Polygons and Circles Triangles: Classification, Properties, and Theorems Triangles are polygons with three sides and three angles. Classification of Triangles: 1. Scalene Triangle: A triangle with no congruent sides. 2. Isosceles Triangle: A triangle with at least two congruent sides. 3. Equilateral Triangle: A triangle with all three sides congruent. Properties of Triangles: - Interior Angles: The sum of the interior angles of a triangle is always 180 degrees. - Exterior Angles: The sum of the exterior angles of a triangle is always 360 degrees. Theorems: - Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Quadrilaterals: Classification, Properties, and Theorems Quadrilaterals are polygons with four sides and four angles. Classification of Quadrilaterals: 1. Parallelogram: A quadrilateral with opposite sides parallel. 2. Rectangle: A parallelogram with four right angles. 3. Rhombus: A parallelogram with all four sides congruent. 4. Square: A rectangle with all four sides congruent. Properties of Quadrilaterals: - Diagonals: The diagonals of a parallelogram bisect each other. - Opposite Angles: Opposite angles in a parallelogram are congruent. Theorems: - The sum of the measures of the interior angles of any quadrilateral is always 360 degrees. Circles: Definitions, Properties, and Formulas Circles: Definitions, Properties, and Formulas A circle is a set of points in a plane that are equidistant from a fixed point called the center. Properties of Circles: - Radius: The distance from the center of a circle to any point on the circle. - Diameter: A line segment that passes through the center of the circle and has endpoints on the circle. - Circumference: The distance around the circle, given by the formula \(C = 2\pi r\), where \(r\) is the radius. - Area: The space enclosed by the circle, given by the formula \(A = \pi r^2\), where \(r\) is the radius. 7: Algebraic Equations (Practice Problems) Section 6: Practice Problems Problem 1: Solve the equation \(2x + 5 = 17\) for \(x\) Solution: Given: \(2x + 5 = 17\) Subtract 5 from both sides: \(2x = 17 - 5 = 12\) Divide both sides by 2: \(x = \frac{12}{2} = 6\) So, the solution is \(x = 6\). Problem 2: Factor the quadratic expression \(x^2 - 4x + 4\) Solution: Given: \(x^2 - 4x + 4\) This quadratic expression is a perfect square trinomial: \((x - 2)^2\). So, the factored form is \((x - 2)^2\). Problem 3: Solve the system of equations: 1. \(3x + 2y = 8\) 2. \(2x - y = 4\) Solution: Solve equation 2 for \(y\): \(y = 2x - 4\). Substitute this expression into equation 1: \(3x + 2(2x - 4) = 8\). Solve for \(x\) and then find \(y\). 8: Geometry (Practice Problems) Section 7: Practice Problems A selection of geometry problems with solutions for practice Problem 1: Solve the equation \(2x + 5 = 17\) for \(x\). Solution: Given: 2x + 5 = 17 Subtract 5 from both sides: 2x = 17 - 5 = 12 Divide both sides by 2: x = 12 / 2 = 6 So, the solution is x = 6.