Introduction to Algebra. The Language Of Mathematics. IntroductionAlgebra is a branch of mathematics that deals with symbols and their relationships. It's often called the "language of mathematics" because it uses symbols to represent unknown quantities and relationships between them.Key Concepts:Variables: These are symbols, usually letters like [latex]x, y[/latex], or [latex]z[/latex], that represent unknown quantities.Expressions: Combinations of variables, numbers, and mathematical operations (addition, subtraction, multiplication, division). For example, [latex]2x + 5[/latex] is an expression.Equations: Statements that two expressions are equal. For example, [latex]2x + 5 = 13[/latex] is an equation.Solving Equations: Finding the value of the variable that makes the equation true.Why is Algebra Important?Problem-Solving: Algebra helps you develop problem-solving skills by teaching you how to break down complex problems into smaller, manageable steps.Real-World Applications: Algebra is used in various fields, including science, engineering, finance, and economics, to model and solve real-world problems.Foundation for Higher Mathematics: Algebra is a foundational subject that prepares you for more advanced math courses like calculus, trigonometry, and statistics.Example:Solve the equation: [latex]2x + 3 = 9[/latex]Step 1: Subtract 3 from both sides to isolate the variable: [latex]2x = 6[/latex]Step 2: Divide both sides by 2 to find the value of [latex]x: x = 3[/latex]Topics in Algebra includeLinear Equations: Equations where the highest power of the variable is 1.Solving a Linear Equation [latex]3x + 5 = 17[/latex] Solution: Subtract 5 from both sides: [latex]3x = 12[/latex] Divide both sides by 3. [latex]frac{3x}{3}=frac{12}{3}[/latex] [latex]x = 4[/latex] Systems of Equation: A set of two or more equations with the same variables. Solving a System of Linear Equations Question: Solve the system of equations: [latex]2x + y = 7[/latex] [latex]x - y = 1[/latex] Solution: Let's solve the above system of equations using the substitution method, [latex]2x + y = 7[/latex] …. eqn(1) [latex]x - y = 1[/latex] …. eqn (2) From eqn (2), [latex]x - y = 1[/latex], [latex]x=1+y[/latex] … eqn (3) Substitute [latex]x=1+y[/latex] into eqn (1) [latex]2x + y = 7[/latex] [latex]2(1+y)+y=7[/latex] [latex]2+2y+y=7[/latex] [latex]2+3y=7[/latex] [latex]3y=7-2[/latex] [latex]3y=5[/latex] [latex]frac{3y}{3}=frac{5}{3}[/latex] [latex]y=1.6667=frac{5}{3}[/latex] Substitute [latex]y=1.6667= frac{5}{3}[/latex] for y into eqn 3 to get [latex]x[/latex]. [latex]x=1+y[/latex] [latex]x=1+1.6667[/latex] [latex]x=2.6667[/latex] Quadratic Equations: Equations where the highest power of the variable is 2. Solving a Quadratic Equation Question: Solve the quadratic equation [latex]x^2 - 5x + 6 = 0[/latex]. Solution: Multiplying 1st and last term; [latex]x^2[/latex] and +6 =[latex]+6x^2[/latex] Factors of =[latex]+6x^2[/latex], such that their sum is [latex]-5x[/latex](the Middle term)[latex]=-3 and -2[/latex]. Now replace the middle term with these new factors. [latex]x^2 - 3x-2x + 6 = 0[/latex] [latex](x^2 - 3x)-(2x + 6) = 0[/latex] Group in 2’s. [latex]x(x - 3)-2(x - 3) = 0[/latex]. Factor out the GCF. Factor the equation: [latex](x - 2)(x - 3) = 0[/latex] Set each factor equal to zero and solve: [latex]x - 2 = 0[/latex] or [latex]x - 3 = 0[/latex] [latex]x= 2[/latex] or [latex]x= 3[/latex] Polynomials: Expressions with multiple terms involving variables and constants. Simplifying an Expression Question: Simplify the expression: [latex]2x^{3} - 3x^{2} + 5x - 1 + x^{3} + 2x^{2} - 3x + 4[/latex] Solution: Combine like terms: [latex]2x^{3} + x^{3} -3x^{2} + 2x^{2}+ 5x - 3x -1 + 4[/latex] Simplified expression: [latex]3x^{3} - x^{2} + 2x + 3[/latex] Graphing: Visualizing equations on a coordinate plane.Graphing a Linear EquationQuestion: Graph the equation [latex]y = 2x + 3[/latex].Solution:Find the y-intercept(The y-intercept of a graph is the point where the graph intersects the y-axis): When x = 0, y = 3. So, the y-intercept is (0, 3).Find the slope: The slope is 2, which means that for every 1 unit increase in x, y increases by 2 units.Plot the y-intercept and use the slope to find another point: Starting from (0, 3), move 1 unit to the right and 2 units up. This gives you the point (1, 5).Draw a line through the two points: This line represents the equation y = 2x + 3.Conclusion: By mastering these concepts, you'll be well-prepared to tackle more advanced algebraic problems and applications.Exercise 12.0Solve for [latex]y: 3y-7 = 11[/latex]Find [latex]x: 2(x + 3)-4 = 12[/latex][latex]3(y-5) + 7 = 16[/latex][latex]frac{1}{2x} + 3 = 7[/latex]Solve the system of equations: [latex]2x + y = 7, x-3y =-4[/latex][latex]3x-2y = 1, 4x + 3y = 8[/latex][latex]x + y = 5, x-y = 1[/latex]Graph the equation: [latex]y = 2x + 3[/latex]Graph the equation: [latex]y =-3x + 1[/latex]Graph the equation: [latex]y = frac{1}{2x}-4[/latex]Submit Answers via Chat e.g Exercise 12.o, then type Answers (Number your Answers).FOR MORE CONTENT!!!Register HERE Now! Pick a course, watch our Videos and take our CBT's.Additional Resources:N/A: https://www.waokmath.com[Image Source]: https://www.freepik.com/.https://www.desmos.com/calculator/0sbpkqpbwr
