Algebraic Expressions and Equations

1. The Language of Algebra

Before we can solve problems, we first need to understand the language used to express them. Algebra extends the arithmetic you're familiar with by introducing the concept of unknown values.

To get a feel for what algebra is all about, let's start with a short, intuitive video.

As the video explained, algebra uses variables (like $x$, $y$, $a$) as placeholders for numbers we don't know yet. These variables are combined with known numbers using arithmetic operations to form algebraic expressions.

An algebraic expression is made up of terms, which are separated by addition or subtraction. For example, in the expression $5x-3y+7$:

• $5x$, $-3y$, and $7$ are the terms.
• In the term $5x$, the number $5$ is the coefficient and $x$ is the variable.
• The term $7$ is a constant because it has no variable part.

2. Manipulating Expressions: Combining Like Terms

Often, our first step in solving a problem is to simplify a complex expression. The most common way to do this is by combining like terms. "Like terms" are terms that have the exact same variable part (including exponents). For example, $5x$ and $7x$ are like terms, but $5x$ and $5x^2$ are not.

This next video demonstrates how to identify and combine like terms, and also introduces a crucial rule for dealing with expressions in parentheses: the distributive property.

To summarize the key idea: to simplify an expression, you first use the distributive property to eliminate any parentheses, and then you combine all the like terms.

For example, to simplify $3(x+2)-(x-1)$:

1. Distribute:
   • $3\cdot x+3\cdot 2=3x+6$
   • $-1\cdot x-1\cdot (-1)=-x+1$
   • The expression becomes: $3x+6-x+1$
2. Combine like terms:
   • Combine the $x$ terms: $3x-x=2x$
   • Combine the constants: $6+1=7$
3. Result: $2x+7$

Now that we can confidently manipulate expressions, we can use this skill to solve equations.

3. Solving Linear and Quadratic Equations

An equation is a statement that two expressions are equal. Our goal when "solving" an equation is to find the value(s) of the variable that make this statement true. Today, we'll focus on two types: linear and quadratic equations.

3.1. Linear Equations

A linear equation with one variable can be written in the standard form $Ax+B=0$, where $A$ and $B$ are constants and $A\ne 0$. Solving it involves one main principle: whatever you do to one side of the equation, you must also do to the other to maintain the balance. The goal is to isolate the variable $x$.

Let's walk through an example. Solve the equation $5x-3=2x+9$.

1. Gather variable terms on one side. To remove $2x$ from the right side, subtract $2x$ from both sides: $$5x-2x-3=2x-2x+9$$ $$3x-3=9$$
2. Gather constant terms on the other side. To remove $-3$ from the left side, add $3$ to both sides: $$3x-3+3=9+3$$ $$3x=12$$
3. Isolate the variable. To get $x$ by itself, divide both sides by its coefficient, 3: $$\frac{3x}{3}=\frac{12}{3}$$ $$x=4$$

The solution is $x=4$. A linear equation in one variable will have exactly one solution.

3.2. Quadratic Equations

Quadratic equations are of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants and $a\ne 0$. These are a bit more complex and can have zero, one, or two solutions. There are three primary methods for solving them.

The following reading from the Australian Mathematical Sciences Institute (AMSI) provides a comprehensive guide to these methods. We will go through it section by section.

Method 1: Factoring
This method is fast if you can easily find the factors. It relies on rewriting the equation $a{x}^2+bx+c=0$ into the form $(px+q)(rx+s)=0$. The solutions are then found by setting each factor to zero.

• Example: $x^2-5x+6=0$
• Factor: $(x-2)(x-3)=0$
• Set each factor to zero: $x-2=0$ or $x-3=0$
• Solutions: $x=2$ or $x=3$

Factoring isn't always easy or even possible with real numbers. For those cases, we need a more robust method.

Method 2: Completing the Square
This method transforms the equation into the form $(x+k{)}^2=d$, which can be solved by taking the square root of both sides.

• Example: $x^2+6x-7=0$

1. Move the constant: $x^2+6x=7$
2. "Complete the square": Take half the coefficient of $x$ ($6/2=3$), square it ($3^2=9$), and add it to both sides: $$x^2+6x+9=7+9$$
3. Factor the left side and simplify the right: $$(x+3{)}^2=16$$
4. Take the square root of both sides (remembering both positive and negative roots): $$x+3=\pm \sqrt{16}⟹x+3=4\text{ or }x+3=-4$$
5. Solve for $x$: $x=1$ or $x=-7$

This method is systematic and leads directly to the most powerful tool for solving quadratics: the quadratic formula.

Method 3: The Quadratic Formula
For any equation in the form $a{x}^2+bx+c=0$, the solutions are given by the formula:

You simply identify the values of $a$, $b$, and $c$ from your equation and substitute them into the formula.

The part under the square root, $D=b^2-4ac$, is called the discriminant. It quickly tells you how many real solutions the equation has:

• If $D>0$, there are two distinct real solutions.
• If $D=0$, there is exactly one real solution.
• If $D<0$, there are no real solutions (the solutions involve imaginary numbers, which are outside the scope of this lesson).

Conclusion

In this foundational lesson, we have covered the essential skills for working with basic algebraic equations. Here are the key takeaways:

• Algebraic expressions use variables to represent unknown values, and can be simplified by combining like terms and using the distributive property.
• Linear equations ($ax+b=0$) involve a variable to the first power and have one unique solution, found by isolating the variable.
• Quadratic equations ($a{x}^2+bx+c=0$) involve a variable to the second power and can have zero, one, or two real solutions.
• They can be solved by factoring (if simple), completing the square (a general method), or using the quadratic formula (a universal tool).

You have taken a significant first step. These algebraic tools are not just abstract math problems; they are the instruments used to model and solve problems across all scientific and engineering disciplines.

In the next lesson, we will build on this foundation by exploring the concept of functions. We'll move from finding a single unknown value to describing the relationship between two or more variables, which is a central concept in programming and modeling systems.