Linear Equations: Chapter Review

Vocabulary and notation

Equation and solution

An equation claims two expressions are equal. A solution is a value making that claim true, confirmed by substituting it into the original equation.

Linear equation in one variable, standard form $ax + b = c$

The variable appears only to the first power: no square, no root, no variable denominator. With $a \neq 0$ there is exactly one solution.

Equivalent equations

Two equations with the same solution set. Each legal move produces one, so solving is a chain of them.

Inverse operations, undone in reverse order

Addition and subtraction undo each other, as do multiplication and division. Strip what wraps the variable in the reverse of the order that built it.

Least common denominator (LCD)

The smallest expression every denominator divides into evenly. Multiplying both sides by it clears all the fractions at once, and when it is a plain number that step reverses, so the solution set is untouched.

Proportion $\dfrac{a}{b} = \dfrac{c}{d}$

An equation stating that two ratios are equal, with $b \neq 0$ and $d \neq 0$.

Excluded value

A value making a denominator zero, so the equation is undefined there.

Contradiction

An equation true for no value of the variable; simplifying leaves a false statement such as $2 = 5$.

Identity

An equation true for every value of the variable; simplifying leaves a true statement such as $4 = 4$, so the solution is all real numbers.

Literal equation

An equation with more than one letter, such as $A = lw$ or $d = rt$. Most formulas are literal equations.

Solving for a variable

Rewriting a literal equation so one chosen letter stands alone, every other letter treated as a fixed but unknown number.

Consecutive integers

Integers stepping by $1$: $n$, $n + 1$, $n + 2$. Consecutive even and consecutive odd integers both step by $2$: $n$, $n + 2$, $n + 4$.

Formulas and theorems

• Properties of equality

  $$\begin{gathered} \text{If } a = b, \text{ then:} \\ a + c = b + c \\ a - c = b - c \\ ac = bc \\ \frac{a}{c} = \frac{b}{c} \end{gathered}$$

  Text description

  A level balance beam whose left pan holds a plus c and whose right pan holds b plus c.

  Use when Any $c$ for adding and subtracting; $c \neq 0$ for multiplying and dividing, since multiplying by $0$ makes any equation $0 = 0$ and cannot be undone. The word number is load-bearing: an expression holding the variable is not one, because it can be zero.

• Solution of the standard form

  $$ax + b = c \quad\Longrightarrow\quad x = \frac{c - b}{a}$$

  Use when $a \neq 0$. Two moves: undo the constant, then the coefficient.

  e.g. $5x + 6 = 26$ gives $x = \frac{26 - 6}{5} = 4$.

• How many solutions

  Gather variable terms on one side and constants on the other, leaving $ax = b$. If $a \neq 0$: one solution, $x = \frac{b}{a}$. If $a = 0$ and $b \neq 0$: none. If $a = 0$ and $b = 0$: infinitely many, every real number.

  Use when The verdict belongs to the equation you cleared TO, and it is the original's too when every multiplier was a nonzero number. Clear with something holding the variable and the survivor still faces the excluded-value check.

  e.g. $4x + 2 = 4x + 5$ collapses to $2 = 5$, false, so no solution.

• Clearing fractions with the LCD

  Multiply both sides by the LCD and distribute it to every term, including terms carrying no fraction; each denominator divides the LCD exactly.

  Use when When the LCD is a plain number the step reverses, so the solution set is unchanged. When it holds the variable, record every value making it zero as excluded and test each candidate against those exclusions. Keep a numerator that is a sum in parentheses.

• Clearing decimals with a power of ten

  Multiply both sides by the power of ten that makes every coefficient a whole number.

  Use when Use $10^k$ for the largest number of decimal places present; a power of ten is nonzero, so the solution is untouched.

  e.g. $0.5x - 0.2 = 0.1x + 1$ times $10$ gives $5x - 2 = x + 10$, so $x = 3$.

• Cross-multiplication

  $$\frac{a}{b} = \frac{c}{d} \quad\Longleftrightarrow\quad ad = bc$$

  Use when $b \neq 0$ and $d \neq 0$; the move is multiplying both sides by $bd$.

  e.g. $\frac{x}{6} = \frac{x+4}{9}$ gives $9x = 6(x + 4)$, so $x = 8$.

• Excluded values

  Set each denominator holding the variable equal to zero; those values are excluded. Solve, then reject any answer equal to one.

  Text description

  A number line from zero to five with the point at three punched out and marked excluded.

  Use when Required whenever the variable sits in a denominator: clearing it multiplies by a quantity that may be zero, and that step does not undo.

  e.g. $\frac{x}{x-3} = \frac{3}{x-3}$ clears to $x = 3$, the excluded value, so there is no solution.

• Ages at another time

  In $k$ years every age is $k$ larger; $k$ years ago every age was $k$ smaller.

  Use when Name one person's present age, write the others' present ages from the given relationship, then shift the whole cast by the same $k$.

  e.g. Maria is $4b$ and her brother $b$; in $6$ years $4b + 6 = 2(b + 6)$, so $b = 3$.

• Distance, rate, time

  $$d = rt$$

  Use when A constant rate, units matched. Rearranged, $r = \frac{d}{t}$ needs $t \neq 0$ and $t = \frac{d}{r}$ needs $r \neq 0$.

  e.g. Riders at $12$ and $16$ mph in opposite directions: $12t + 16t = 70$ miles apart at $t = 2.5$ hours.

• Total value of a collection

  $$\begin{gathered} \text{total value} = \\ (\text{number of items}) \\ \times\ (\text{value of each}) \end{gathered}$$

  Use when Items identical within a kind and all values in one unit; use cents for coins. Count one kind with the variable, the other from the total.

  e.g. $30$ coins, nickels and dimes, worth $245$ cents: $5n + 10(30 - n) = 245$, so $n = 11$ nickels.

• Percent translation

  $p\%$ of a quantity is $\frac{p}{100}$ times it; "of" multiplies and "is" is an equals sign. A $p\%$ discount leaves $(100 - p)\%$ of the price.

  Text description

  A price bar split into a p percent discount and the 100 minus p percent you actually pay.

  Use when Convert to a decimal before solving; the discount form assumes $0 \le p \le 100$. Confirm the answer makes sense as a count or a price.

  e.g. $34$ is $85\%$ of $t$ means $0.85t = 34$, so $t = 40$.

Problem types, step by step

Exam traps

• Trap Multiplying only the fractions by the LCD and leaving a whole-number term alone.

  Fix The multiplier hits every term on both sides: $\frac{x}{2} + \frac{x}{3} = 5$ times $6$ is $3x + 2x = 30$, not $3x + 2x = 5$.

• Trap Letting a minus sign reach only the first term of a grouped numerator or parenthesis.

  Fix A fraction bar groups its numerator, so $-\frac{x-3}{5}$ times $10$ is $-2(x - 3) = -2x + 6$, not $-2x - 6$; likewise $-(x - 4) = -x + 4$.

• Trap Cross-multiplying when a side is a sum of fractions or carries an extra term.

  Fix Cross-multiplication belongs to $\frac{a}{b} = \frac{c}{d}$ alone. Anything else: clear with the LCD first, then solve.

• Trap Reading a cancelled variable as $x = 0$.

  Fix A false leftover such as $2 = 5$ means no solution; a true one such as $4 = 4$ means every real number is a solution. Neither is a value of $x$.

• Trap Trusting the one, none, or infinitely many verdict after clearing a denominator that held the variable.

  Fix Clearing with a number reverses, so the verdict transfers. Clearing with an expression such as $x - 3$ may multiply by zero, which is why $\frac{x}{x-3} = \frac{3}{x-3}$ has no solution.

• Trap Dividing only the first term of a multi-term side by the coefficient.

  Fix $2w = P - 2l$ gives $w = \frac{P - 2l}{2}$, which is $\frac{P}{2} - l$, never $\frac{P}{2} - 2l$. Every term crosses the division.

• Trap Factoring the target out and losing the invisible $1$.

  Fix The lone $P$ is $P \cdot 1$, so $A = P + Prt$ is $A = P(1 + rt)$ and $P = \frac{A}{1 + rt}$, never $\frac{A}{rt}$. Expand any factoring back to check it.

• Trap Translating "less than" and "more than" in the order the words arrive.

  Fix The amount taken away comes second: "$5$ less than $x$" is $x - 5$, not $5 - x$; "$3$ less than twice a number" is $2n - 3$. Addition is order-blind, which is why only the subtraction reverses.

• Trap Applying a move to one side only, such as subtracting $6$ from the left of $2x + 6 = 20$ and leaving the right untouched.

  Fix That leaves $2x = 20$ and the wrong answer $x = 10$; subtracting $6$ from both sides leaves $2x = 14$, so $x = 7$. Every move must land on both sides at once, or the new equation is no longer equivalent to the old one.