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Algebra I · Lesson 1.4

Expressions

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An expression is a set of instructions for doing a calculation. When an expression contains a letter (called a variable), that letter can represent any number. Because of this, a single algebraic expression works for infinitely many cases instead of just one.

Problem
A print shop charges the same way no matter how many posters you order. One poster costs $12, two cost $19, and three cost $26. Each extra poster adds the same amount, and there is a one-time setup charge baked into every order. What does an order of 10 posters cost, in dollars?
Show a hint
  • From 1 poster to 2, and from 2 to 3, the price climbs by the same fixed step. Find that step first.
  • Once you know the per-poster step, work backward to the setup charge hiding inside the price of one poster.
Show the full solution
Each new poster adds 19−12=7 dollars, so ten posters cost 70 for printing. One poster costs 12, which is 7 for the poster plus 5 of setup, so the setup is 5 and every order is 5+7p. At p=10 that is 5+70=75.
Problem
An expression splits into pieces wherever a + or a − sign falls between them, and each piece is called a term. Someone writes 8−3x+5xy+y=4. Counting only what sits to the left of the equals sign, how many terms does that expression have?
Show a hint
  • Slice the expression at every + and every −. Each slice is one term, and the leading 8 counts.
Show the full solution
Cutting at each + and − gives 8, −3x, 5xy, and y, which is 4 terms. The minus stays on the 3x, and the leading 8 counts as a term even though nothing sits in front of it.
Problem
The coefficient of a term is its numerical factor, and it carries the sign sitting in front of it. In the expression 10−6w, what is the coefficient of w?
Show a hint
  • Subtracting 6w is adding −6w, so the sign in front travels with the number multiplying w.
Show the full solution
The term holding w is −6w, so its numerical factor is −6. The minus in front belongs to the term, so answering 6 drops the sign.
Problem
Two expressions look almost the same but do very different things. Writing 4n means 4 times n, while 4+n means 4 plus n. When n=3, how much larger is 4n than 4+n?
Show a hint
  • 4n is a product and 4+n is a sum. Evaluate each at n=3 before comparing.
  • The little gap between the 4 and the n is doing all the work. No sign there means multiply, a plus sign means add.
Show the full solution
At n=3, the product 4n=4⋅3=12 and the sum 4+n=4+3=7, so the product runs ahead by 12−7=5. The gap between the 4 and the n means multiply, which is easy to read as a plus.
Plus and minus signs cut the expression into terms13+24x-17xytermtermterm, minus includedMultiplication joins factors inside one term24x24coefficient·xvariable factor
One expression, cut two ways. On top the plus and minus signs split 13+24x−17xy into its three terms, and the minus stays with the term it marks. Below, the term 24x opens into the two factors multiplied inside it, the coefficient 24 and the variable factor x. Plus and minus separate terms, multiplication joins factors.
Problem
At a fair, each ride costs $3 and each game costs $5. A single expression, 3r+5g, gives the total spend for r rides and g games. What is the total, in dollars, for 4 rides and 6 games?
Show a hint
  • The 3r counts all the ride money and the 5g counts all the game money. Find each, then add.
Show the full solution
The rides cost 3r=3⋅4=12 dollars and the games cost 5g=5⋅6=30 dollars, for a total of 12+30=42. Each rate stays with the count it belongs to, so the 3 never multiplies the games.
Problem
Maria writes the total for a party as 6(t+2), while Jon writes it as 6t+12. They insist both give the same number for the same t. For t=8, what value do both expressions produce?
Show a hint
  • Evaluate each expression on its own at t=8, and do the grouped one inside first. Do not rewrite one into the other.
Show the full solution
Maria's 6(t+2)=6⋅10=60, and Jon's 6t+12=48+12=60, so both give 60. Matching at one value is a good sign but not proof, since two expressions can agree at one t and split at the next.
Problem
It is tempting to treat 3(n+4) and 3n+4 as the same expression, but they are not. At n=2, by how much do their values differ?
Show a hint
  • Evaluate both at n=2, the grouped one first, then subtract the smaller from the larger.
  • One value that disagrees is enough to prove two expressions are not the same.
Show the full solution
At n=2, 3(n+4)=3⋅6=18 while 3n+4=6+4=10, so they differ by 8. One disagreeing value is enough to prove two expressions are not the same, and that gap of 8 shows up at every n.
Problem
A parking garage charges $6 to enter plus $4 for every hour parked, but the first hour is validated free, so you pay for one fewer hour than you stay. The charge for staying h hours is 6+4(h−1) dollars. What do you pay, in dollars, for staying 9 hours?
Show a hint
  • The 6 is a flat entry charge that is not multiplied by anything. The 4(h−1) is the hourly charge on the hours you actually pay for.
  • With one hour free, staying 9 hours means paying for 8 of them.
Show the full solution
Nine hours means paying for h−1=8 of them, so the hourly part is 4⋅8=32 dollars, and the flat 6 is added on top for 6+32=38. The 6 sits outside the multiplication, so it never gets multiplied by the hours.

This lesson added seven words to your working vocabulary. A variable is a letter that represents a number. An expression is a set of instructions that may use variables. Its terms are the pieces split by + and −, and each term's coefficient is its signed numeric factor. Factors are what multiply inside a term. A constant is a term with no variable. Equivalent expressions agree at every value. Next, 1.5 turns to exponents, the shorthand for repeated multiplication.

Practice these ideas

Practice
Terms are the pieces of an expression separated by + and − signs. How many terms are in 4a+3−b?
Show the solution
The pieces are 4a, 3, and −b, so there are 3.
Practice
In 7−4xy, what is the coefficient of the term xy, sign included?
Show the solution
The term is −4xy, so its numerical factor, sign and all, is −4. The minus belongs to the term, which makes it part of the coefficient too.
Practice
A taxi charges $4 to start plus $2 for each mile, so a ride of m miles costs 4+2m dollars. What does an 8-mile ride cost, in dollars?
Show the solution
The mileage part is 2⋅8=16 dollars on top of the flat 4, so the cost is 4+16=20.
Practice
A theater fills its rows by a fixed rule. Row 1 seats 6, row 2 seats 10, and row 3 seats 14, each row adding the same number of seats. How many seats are in row 20?
Show the solution
Each row adds 4 seats, so row r seats 2+4r, since row 1 gives 2+4=6. Row 20 seats 2+4⋅20=82.
Practice
The constant term of an expression is the term with no variable. What is the constant term of 9x−5+2x2?
Show the solution
Only −5 has no variable, and the sign stays with it, so the constant term is −5.
Practice
Recall that 3n means 3 times n, while 3+n means 3 plus n. At n=5, how much larger is 3n than 3+n?
Show the solution
At n=5, the product 3n=15 and the sum 3+n=8, so the product leads by 15−8=7.
Practice
The expressions 4(n+3) and 4n+3 are not equal. At n=1, by how much do their values differ?
Show the solution
At n=1, 4(n+3)=4⋅4=16 while 4n+3=4+3=7, so they differ by 9. One disagreeing value proves two expressions are not the same, and this gap of 9 holds at every n.
Practice
The expression 5a+3b counts two things at once. Find its value when a=7 and b=3.
Show the solution
Here 5a=35 and 3b=9, so 5a+3b=35+9=44.
Practice
Evaluate 10−2(m+1) when m=3, keeping the order of operations in mind.
Show the solution
Inside the parentheses, m+1=4, then 2⋅4=8, and 10−8=2. Subtracting the 2 from the 10 first would ignore the grouping and give the wrong value.
Practice
Three expressions sit side by side, 2(n+3), 2n+3, and 2n+6. Two of them are equivalent, giving the same value for every n, and one is the odd one out. Test all three at n=4, and enter the value that the two equivalent ones share.
Show the solution
At n=4 the three values are 2(4+3)=14, 2⋅4+3=11, and 2⋅4+6=14, so the shared value is 14. The odd one out is 2n+3, while 2(n+3) and 2n+6 agree at every n, as Chapter 2 will show.