Algebra is the language that lets you describe a pattern once and use it again and again. In this chapter you will learn to read and write algebraic expressions, to compute with them in the right order, and to rewrite them in simpler forms using the properties of real numbers. These tools are the foundation of everything else in Algebra 1.
1. Variables and algebraic expressions
A variable is a letter that stands for a number that can change or that you do not know yet. A constant is a number that never changes. An algebraic expression combines numbers, variables, and operation symbols, such as \(5n+12\) or \(3x^2-7x+4\). An expression has no equal sign: it is a phrase, not a complete sentence.
The parts separated by plus or minus signs are called terms. In \(3x^2-7x+4\) the terms are \(3x^2\), \(-7x\), and \(4\). The number multiplying a variable is its coefficient: here \(3\) and \(-7\). The term with no variable, \(4\), is the constant term.
Remember that a sign in front of a term belongs to that term. Also, \(5n\) means \(5\cdot n\), and \(n\) alone has the coefficient \(1\), because \(n=1\cdot n\).
2. Order of operations
When an expression has several operations, everyone must compute in the same order, or the same expression would have different values. The agreed order is often remembered as PEMDAS.
- Work inside parentheses and brackets first, starting with the innermost pair.
- Compute exponents (powers) next.
- Do multiplications and divisions from left to right, as they appear.
- Do additions and subtractions from left to right.
Compute \(18-4\cdot(2+1)^2\div 6\).
Parentheses: \(2+1=3\). Exponent: \(3^2=9\). Multiplication and division from left to right: \(4\cdot 9=36\), then \(36\div 6=6\). Subtraction: \(18-6=12\).
So \(18-4\cdot(2+1)^2\div 6=12\).
Multiplication does not always come before division, and addition does not always come before subtraction. They share the same rank, so you go left to right: \(24\div 4\cdot 3=6\cdot 3=18\), not \(24\div 12=2\). Also, \(-3^2=-9\), but \((-3)^2=9\): the exponent only applies to what is right next to it.
3. Evaluating expressions
To evaluate an expression means to replace each variable by a given number and then compute. Always put the replacement number in parentheses, especially when it is negative, so that signs and exponents are handled correctly.
Evaluate \(3x^2-2xy+y\) for \(x=-2\) and \(y=5\).
Substitute: \(3(-2)^2-2(-2)(5)+5\). Exponent first: \(3\cdot 4-2(-2)(5)+5\). Multiply: \(12+20+5\). Add: \(37\).
The value is \(37\).
A weather app converts Celsius to Fahrenheit with the expression \(1.8C+32\). When it is \(25^{\circ}\text{C}\) outside, we get \(1.8(25)+32=45+32=77\). So it is \(77^{\circ}\text{F}\).
4. Properties of real numbers
Properties are rules that are always true for any real numbers. They explain why certain rewritings are allowed, and they make mental math and algebra much faster.
| Property | In symbols | Example |
|---|---|---|
| Commutative (addition) | \(a+b=b+a\) | \(7+x=x+7\) |
| Commutative (multiplication) | \(a\cdot b=b\cdot a\) | \(6\cdot 9=9\cdot 6\) |
| Associative (addition) | \((a+b)+c=a+(b+c)\) | \((4+5)+1=4+(5+1)\) |
| Associative (multiplication) | \((ab)c=a(bc)\) | \((2\cdot 7)\cdot 5=2\cdot(7\cdot 5)\) |
| Distributive | \(a(b+c)=ab+ac\) | \(3(x+4)=3x+12\) |
| Identities | \(a+0=a\), \(a\cdot 1=a\) | \(8+0=8\), \(1\cdot y=y\) |
| Inverses | \(a+(-a)=0\), \(a\cdot\dfrac{1}{a}=1\ (a\neq 0)\) | \(5+(-5)=0\), \(5\cdot\dfrac{1}{5}=1\) |
The commutative properties say that order does not matter for adding or multiplying. The associative properties say that grouping does not matter. The distributive property connects multiplication and addition, and it is the most used property in algebra. An area picture shows why it works.
\(5-2\neq 2-5\) and \(12\div(3\cdot 2)\neq(12\div 3)\cdot 2\) because \(12\div 6=2\) but \(4\cdot 2=8\). To get around this, rewrite subtraction as adding the opposite, \(a-b=a+(-b)\), and division as multiplying by the reciprocal.
Compute \(25\cdot 17\cdot 4\) quickly. By the commutative and associative properties, \(25\cdot 17\cdot 4=(25\cdot 4)\cdot 17=100\cdot 17=1{,}700\). Similarly, \(8\cdot 53=8(50+3)=400+24=424\).
5. Combining like terms
Like terms have exactly the same variable part, with the same exponents. Only the coefficients can differ. So \(4x\) and \(-9x\) are like terms, and \(6xy^2\) and \(xy^2\) are like terms, but \(3x\) and \(3x^2\) are not, and neither are \(2a\) and \(2b\). Constants are like terms with each other.
The distributive property lets you add the coefficients: \(4x+9x=(4+9)x=13x\). The variable part does not change.
Simplify \(5x^2+3x-2x^2+7-x+4\).
Group by type: \((5x^2-2x^2)+(3x-x)+(7+4)\). Add the coefficients: \(3x^2+2x+11\).
6. Simplifying expressions
An expression is simplified when it has no parentheses and no like terms left to combine. To get there, distribute first, then combine like terms.
- Distribute every factor or negative sign that sits in front of parentheses, multiplying it by every term inside.
- Underline or circle terms of the same type, keeping their signs.
- Add the coefficients of the like terms.
- Write the result, usually with the highest power first.
Simplify \(4(2x-3)-3(x-5)\).
Distribute: \(8x-12-3x+15\). Notice that \(-3\cdot(-5)=+15\). Combine: \((8x-3x)+(-12+15)=5x+3\).
Check with \(x=2\): the original gives \(4(1)-3(-3)=4+9=13\), and \(5(2)+3=13\). It matches.
On my planet we always test our work: pick an easy number such as \(x=2\) and evaluate the expression before and after simplifying. If the two results differ, a mistake is hiding somewhere!
7. The real number system
Every number you meet in Algebra 1 is a real number, which means it has a place on the number line. Real numbers are organized into sets that sit inside each other.
- Natural numbers (counting numbers): \(1,2,3,\dots\)
- Whole numbers: \(0,1,2,3,\dots\)
- Integers: \(\dots,-2,-1,0,1,2,\dots\)
- Rational numbers: numbers that can be written as \(\dfrac{a}{b}\) with integers \(a\) and \(b\neq 0\). Their decimals stop or repeat, like \(0.75\) or \(0.\overline{3}\).
- Irrational numbers: real numbers that are not rational. Their decimals never stop and never repeat, like \(\sqrt{2}\approx 1.414\) and \(\pi\approx 3.14159\).
Classify \(\sqrt{49}\), \(-\dfrac{3}{4}\), and \(\sqrt{11}\).
\(\sqrt{49}=7\), which is a natural number, so it is also whole, an integer, rational, and real. \(-\dfrac{3}{4}\) is a fraction of integers, so it is rational and real but not an integer. \(11\) is not a perfect square, so \(\sqrt{11}\) is irrational and real.
The sum or product of two rational numbers is rational. The product of a nonzero rational number and an irrational number is irrational. The sum of two irrational numbers can be rational (for example \(\sqrt{2}+(3-\sqrt{2})=3\)).
8. Translating words into expressions
Many word problems start with turning a phrase into an expression. Look for key words and choose a variable for the unknown number.
| Words | Expression |
|---|---|
| the sum of a number and 9; 9 more than a number | \(n+9\) |
| a number decreased by 4; 4 less than a number; 4 subtracted from a number | \(n-4\) |
| the difference of 4 and a number (the order is reversed) | \(4-n\) |
| 7 times a number; the product of 7 and a number | \(7n\) |
| the quotient of a number and 6 | \(\dfrac{n}{6}\) |
| twice a number, increased by 5 | \(2n+5\) |
“Five less than twice a number” is written \(2n-5\), not \(5-2n\): first “twice a number” gives \(2n\), then we take away \(5\). A taxi that charges a $3 base fee plus $2 per mile costs \(3+2m\) dollars for \(m\) miles; for a 7-mile ride that is \(3+2(7)=17\) dollars.
Key takeaways
- An expression is made of terms; coefficients multiply variables and the constant term has no variable.
- Follow PEMDAS: parentheses, exponents, multiplication and division left to right, addition and subtraction left to right.
- To evaluate, substitute the number in parentheses and compute in order.
- Addition and multiplication are commutative and associative; the distributive law is \(a(b+c)=ab+ac\).
- Only like terms can be combined: add the coefficients and keep the variable part.
- Simplify by distributing first, then combining like terms; check with a test value.
- Real numbers include natural, whole, integer, rational, and irrational numbers.
- “Less than” and “subtracted from” reverse the order of the subtraction.
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