QuanticaAlgebra IOpen in the courseAn expression is a recipe for a computation. A letter in that recipe stands for a number, any number at all. That is what gives algebra its reach, since one recipe with a letter in it handles every number the letter could be, all at once, instead of one arithmetic problem at a time.
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 \(5+70=\boxed{75}\).
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 \(\boxed{4}\) 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 \(\boxed{-6}\). 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 \(4n=4\cdot 3=12\) and the sum \(4+n=4+3=7\), so the product runs ahead by \(12-7=\boxed{5}\). The gap between the \(4\) and the \(n\) means multiply, which is easy to read as a plus.
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 \(3r=3\cdot 4=12\) dollars and the games cost \(5g=5\cdot 6=30\) dollars, for a total of \(12+30=\boxed{42}\). 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 \(6(t+2)=6\cdot 10=60\), and Jon's \(6t+12=48+12=60\), so both give \(\boxed{60}\). 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 \(n=2\), \(3(n+4)=3\cdot 6=18\) while \(3n+4=6+4=10\), so they differ by \(\boxed{8}\). 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 \(4\cdot 8=32\) dollars, and the flat \(6\) is added on top for \(6+32=\boxed{38}\). 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 for a number, an expression is a recipe using them, its terms are the pieces split by \(+\) 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, and 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 \(\boxed{3}\).
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 \(\boxed{-4}\). 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 \(2\cdot 8=16\) dollars on top of the flat \(4\), so the cost is \(4+16=\boxed{20}\).
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 \(2+4\cdot 20=\boxed{82}\).
Practice
The constant term of an expression is the term with no variable. What is the constant term of \(9x-5+2x^2\)?
Show the solution
Only \(-5\) has no variable, and the sign stays with it, so the constant term is \(\boxed{-5}\).
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 \(15-8=\boxed{7}\).
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 \(n=1\), \(4(n+3)=4\cdot 4=16\) while \(4n+3=4+3=7\), so they differ by \(\boxed{9}\). 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 \(5a+3b=35+9=\boxed{44}\).
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 \(2\cdot 4=8\), and \(10-8=\boxed{2}\). 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 \(2(4+3)=14\), \(2\cdot 4+3=11\), and \(2\cdot 4+6=14\), so the shared value is \(\boxed{14}\). The odd one out is \(2n+3\), while \(2(n+3)\) and \(2n+6\) agree at every \(n\), as Chapter 2 will show.