02/14 2. Linear Equations and Applications Free Online FlashCards

What is a linear equation in one variable?

A linear equation in one variable can be written as $ax+b=c$, where $a$, $b$, and $c$ are constants and $a\ne0$. The variable has exponent 1 and is not in a denominator or radical.

What principle justifies doing the same operation to both sides?

Apply the same operation to both sides. Addition, subtraction, multiplication, or division by a nonzero number preserves equality.

Solve $5x-7=2x+11$.

Subtract $2x$, add 7, and divide by 3: $5x-7=2x+11$ becomes $3x=18$, so $x=6$.

Solve $\frac{x}{3}+\frac{1}{2}=\frac{5}{6}$.

Multiply every term by the least common denominator, 6: $2x+3=5$. Thus $2x=2$ and $x=1$.

How many solutions does $2x+5=2x+9$ have?

It has no solution. Simplifying gives $5=9$, a contradiction that is never true.

How many solutions does $4x+8=4(x+2)$ have?

It has infinitely many solutions. Simplifying gives $0=0$, which is true for every value of $x$.

Solve $A=\frac{1}{2}bh$ for $h$.

Multiply by 2 and divide by $b$, assuming $b\ne0$: $h=\frac{2A}{b}$.

How can you solve $\frac{a}{b}=\frac{c}{d}$?

Cross-multiply to obtain $ad=bc$. This is valid when the denominators $b$ and $d$ are nonzero.

What is the basic percent equation?

The part equals the percent written as a decimal times the whole: $\text{part}=\text{percent}\times\text{whole}$.

What formulas model percent increases and decreases?

Use $\text{new amount}=\text{original amount}(1-r)$ for a decrease and $\text{original amount}(1+r)$ for an increase, where $r$ is decimal.

What process solves a linear application problem?

Define the variable, write an equation, solve it, check the result, and state the answer with appropriate units and meaning.

How are three consecutive integers represented?

Let the first integer be $n$; the next two are $n+1$ and $n+2$. Their sum is $n+(n+1)+(n+2)$.