# Foundations for learning Algebra In [[Pre-Algebra]], we saw how [[number|numbers]] interact via [[operation|operations]] to form [[Arithmetic|arithmetic]]. We took complicated [[numerical expressions]] and [[simplify|simplified]] them according to the [[order of operations]] When a certain sequence of operations might be useful with more than just one set of numbers, we can reserve spaces for these numbers. Instead of just leaving a blank space ambiguously at each point where a number would be, we use a [[variable|variable]] represent the places where a number be placed. This results in a [[variable expression]] When the value of the variables are determined, it is possible to [[evaluate variable expressions]]. Before [[substituting]] the known values in for the variables, its helpful to [[simplify variable expressions]] so that the process of evaluating is easier. If two expressions simplify to the same [[simplest form]] then they are equivalent. Two equivalent expressions can look wildly different, so it is good to have ways to describe expressions, often by looking at their parts. In the order of operations, the last operations are [[addition]] and [[subtraction]], so we often cut up expressions using these operations (as long as they are not in grouping symbols or implied groups). These parts are called [[terms]] Before adding and subtracting, in the order of operations, we have multiplication and division, which are the only operations that happen in a term (unless addition or subtraction are hidden in a grouping symbol). Inside of a term, the numerical part of the product is called the [[coefficient]] and the rest are either variables or groups. When a term has no variable part, meaning the term is simply just its coefficient, that term is called a [[constant]]. We categorize other terms based on their variable contents. A term like $7x$ is an $x$ term, while a term like $-13x^2y$ is an $x^2y$ term. If two terms are of the same category, we call them [[like terms]]. We can [[combine like terms]] combining their coefficients, while maintaining their variable part. When an expression has like terms that can be combined, it is not simplified. We can also minimize the need for grouping symbols by applying the surrounding operations if possible. In cases where a parenthesis contains addition or subtraction, any factors that are also part of the same term can be [[distributive property]] to all terms interior to the parenthesis. There will be similar processes for more complicated instances of grouping. With some experimentation, one can be convinced that all expressions utilizing rational number, along with the operations addition, subtraction, multiplication, and division (not by zero), will always have a result that is also a rational number. We will show later that between any two rational numbers (no matter how close) we can always make more rational numbers. In other words, there are infinitely many rational numbers. Still even with their mighty population, the rational numbers do not dominate every spot on the number line. There are numbers that are not part of this set of rational numbers, and they are far more plentiful; we call them [[Irrational number|irrational numbers]]. It might seem contradictory to say that there are infinitely many members of the rational numbers, but then say there are even more irrational numbers; anyone who reads that for the first time asks how there could possibly be any room for more numbers on the number line. For many, this 'paradox' and others like it serve as motivation to progress in mathematics: with enough study you either have enough information to understand or disprove such ludicrous statements. Even so, students in [[Algebra 1]] will continue to see primarily rational numbers, especially integers, more often than irrational numbers. Irrational numbers become more popular when we are solving equations that either use weirder operations, or just infinitely many simple operations. --- The majority of the time in [[Algebra]], the result of a variable expression is known, and the goal is to find the value of the variables. When an expression is known to be equal to something (be it a number or another expression), that statement of equality is an [[equation]], and determining a variable's value so that the equation is true is called [[solving]] This is not unlike a detective that sees the scene of the crime and tries to find the individuals involved. Experience with similar crime-scenes can lead to quicker investigation, so we study some of the most common and useful patterns to make solving equations easier. ## An Introduction to Equations An equation is a statement of equality between two expressions, usually involving one or more variables. In algebra, we use equations to represent and solve real-world problems. A single-variable equation is an equation that contains only one variable, usually represented by a letter such as $x$ or $y$. For example, the equation $3x+2=8$ is a single-variable equation because it only contains the variable $x$. To solve a single-variable equation, we use the goal of isolating the variable on one side of the equation. This is often done by using the [[inverse operations]] of addition and subtraction, multiplication and division, and raising to a power and taking the square root. For example, to solve the equation $3x+2=8$, we can start by subtracting 2 from both sides: $3x+2-2=8-2$ $3x=6$ Next, we can divide both sides by 3: $\frac{3x}{3}=\frac{6}{3}$ $x=2$ Another example is the equation $4x-3=11$. To solve this equation, we can start by adding 3 to both sides: $4x-3+3=11+3$ $4x=14$ Next, we can divide both sides by 4: $\frac{4x}{4}=\frac{14}{4}$ $x=\frac{14}{4}$ $x=3.5$ It is important to note that there may be more than one solution to a single-variable equation, or no solution at all. For example, the equation $x+2=x$ has no solution because the left-hand side and the right-hand side are always equal, regardless of the value of $x$. In conclusion, equations are a fundamental tool in algebra that allow us to represent and solve real-world problems. By using inverse operations and isolating the variable, we can find the solution to a single-variable equation. Practice and perseverance are key to mastering the art of solving equations. Concept Byte: Graphing in the Coordinate Plane 1-9 Patterns, Equations, and Graphs ### Solving Equations #### Using Tables to Solve Equations One way to solve a single-variable equation is by using a table of values. This method can be helpful for visualizing the relationship between the variable and the values of the equation. For example, let's solve the equation $2x+1=5$. | $x$ | $2 x+1$ | $5$ | |---|-------|----| | -2 | -3 | 5 | | -1 | -1 | 5 | | 0 | 1 | 5 | | 1 | 3 | 5 | | 2 | 5 | 5 | To create a table of values, we start by listing the possible values of the variable in the first column. Next, we substitute each value into the equation and evaluate it to find the corresponding value on the right-hand side. By looking at the table, we can see that when $x=2$ the left side of the equation equals to the right side of the equation so, $x=2$ is the solution for this equation. It is important to note that when solving equations with tables, we should always check all possible values of the variable to make sure we have found all solutions. 2-1 Solving One-Step Equations 2-2 Solving Two-Step Equations 2-3 Solving Multi-Step Equations 2-4 Solving Equations with Variables on Both Sides 2-5 Literal Equations and Formulas 2-6 Ratios, Rates, and Conversions 2-7 Solving Proportions 2-8 Proportions and Similar Figures 2-9 Percents 2-10 Change Expressed as a Percent ### Solving Inequalities 3-1 Inequalities and Their Graphs 3-2 Solving Inequalities Using Addition or Subtraction 3-3 Solving Inequalities Using Multiplication or Division Concept Byte: More Algebraic Properties Concept Byte: Modeling Multi-Step Inequalities 3-4 Solving Multi-Step Inequalities 3-5 Working with Sets 3-6 Compound Inequalities 3-7 Absolute Value Equations and Inequalities 3-8 Unions and Intersections of Sets Chapter 4: An Introduction to Functions 4-1 Using Graphs to Relate Two Quantities 4-2 Patterns and Linear Functions 4-3 Patterns and Nonlinear Functions 4-4 Graphing a Function Rule Concept Byte: Graphing Functions and Solving Equations 4-5 Writing a Function Rule 4-6 Formalizing Relations and Functions 4-7 Sequences and Functions Chapter 5: Linear Functions 5-1 Rate of Change and Slope 5-2 Direct Variation Concept Byte: Investigating y= mx + b 5-3 Slope-Intercept Form 5-4 Point-Slope Form 5-5 Standard Form 5-6 Parallel and Perpendicular Lines 5-7 Scatter Plots and Trend Lines Concept Byte: Collecting Linear Data 5-8 Graphing Absolute Value Functions Concept Byte: Characteristics of Absolute Value Graphs Chapter 6: Systems of Equations and Inequalities 6-1 Solving Systems by Graphing Concept Byte: Solving Systems Using Tables and Graphs Concept Byte: Solving Systems with Algebra Tiles 6-2 Solving Systems Using Substitution 6-3 Solving Systems Using Elimination 6-3b Concept Byte: Matrices and Solving Systems 6-4 Applications of Linear Systems 6-5 Linear Inequalities 6-6 Systems of Linear Inequalities Concept Byte: Graphing Linear Inequalities Chapter 7: Exponents and Exponential Functions 7-1 Zero and Negative Exponents 7-2 Scientific Notation 7-3 Multiplying Powers with the Same Base Concept Byte: Powers of Powers and Powers of Products 7-4 More Multiplication Properties of Exponents 7-5 Division Properties of Exponents 7-6 Exponential Functions Concept Byte: Activity: Geometric Sequences 7-7 Exponential Growth and Decay Chapter 8: Polynomials and Factoring 8-1 Adding and Subtracting Polynomials 8-2 Multiplying and Factoring Concept Byte: Activity: Using Models to Multiply 8-3 Multiplying Binomials 8-4 Multiplying Special Cases Concept Byte: Using Models to Factor 8-5 Factoring x^2 + bx + c 8-6 Factoring ax^2 + bx + c 8-7 Factoring Special Cases 8-8 Factoring by Grouping Chapter 9: Quadratic Functions and Equations 9-1 Quadratic Graphs and Their Properties 9-2 Quadratic Functions Concept Byte: Collecting Quadratic Data 9-3 Solving Quadratic Equations Concept Byte: Finding Roots 9-4 Factoring to Solve Quadratic Equations 9-5 Completing the Square 9-6 The Quadratic Formula and the Discriminant 9-7 Linear, Quadratic, and Exponential Models Concept Byte: Performing Regressions 9-8 Systems of Linear and Quadratic Equations Chapter 10: Radical Expressions and Equations 10-1 The Pythagorean Theorem Concept Byte: Distance and Midpoint Formulas 10-2 Simplifying Radicals 10-3 Operations with Radical Expressions 10-4 Solving Radical Equations 10-5 Graphing Square Root Functions Concept Byte: Right Triangle Ratios 10-6 Trigonometric Ratios Chapter 11: Rational Expressions and Functions 11-1 Simplifying Rational Expressions 11-2 Multiplying and Dividing Rational Expressions Concept Byte: Dividing Polynomials Using Algebra Tiles 11-3 Dividing Polynomials 11-4 Adding and Subtracting Rational Expressions 11-5 Solving Rational Equations 11-6 Inverse Variation 11-7 Graphing Rational Equations Concept Byte: Graphing Rational Functions Chapter 12: Data Analysis and Probability 12-1 Organizing Data Using Matrices 12-2 Frequency and Histograms 12-3 Measures of Central Tendency and Dispersion Concept Byte: Standard Deviation 12-4 Box-and-Whisker Plots Concept Byte: Designing Your Own Survey 12-5 Samples and Surveys Concept Byte: Misleading Graphs and Statistics 12-6 Permutations and Combinations 12-7 Theoretical and Experimental Probability Concept Byte: Conducting Simulations 12-8 Probability of Compound Events Conditional Probability