Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the general form of a linear equation in two variables?
Answer
$ax + by + c = 0$, where $a, b,$ and $c$ are real numbers and $x, y$ are variables.
Question
In a linear equation in two variables, what must be the degree of each variable?
Answer
Each variable must have a degree of $1$ (one).
Question
Two linear equations containing the same two variables that are considered together are called _____ equations.
Answer
simultaneous (linear)
Question
What does it mean to 'solve' a system of two simultaneous linear equations?
Answer
To find the specific values of the variables that satisfy both equations at the same time.
Question
What are the three primary algebraic methods discussed for solving simultaneous equations?
Answer
1. Elimination by substitution, 2. Elimination by equating coefficients, and 3. Cross-multiplication.
Question
Method of elimination by substitution: What is the objective of Step 1?
Answer
To express one variable in terms of the other variable using one of the given equations.
Question
Method of elimination by substitution: What is done with the expression obtained in Step 1?
Answer
It is substituted into the other equation to create a single-variable equation.
Question
Method of elimination by substitution: How is the value of the second unknown variable found in the final step?
Answer
By substituting the solved value of the first variable back into the expression from Step 1.
Question
Method of elimination by equating coefficients: What is the first step to prepare the equations?
Answer
Multiply one or both equations by suitable numbers so the coefficients of one variable become numerically equal.
Question
Method of equating coefficients: Under what condition should you subtract one equation from the other?
Answer
When the terms with equal numerical coefficients have the same sign.
Question
Method of equating coefficients: Under what condition should you add the two equations?
Answer
When the terms with equal numerical coefficients have opposite signs.
Question
In equations where the $x$-coefficient of the first is the $y$-coefficient of the second (and vice-versa), what is the first simplifying step?
Answer
Add the two equations together to obtain a simpler equation, usually in the form $x + y = k$.
Question
In equations where the $x$-coefficient of the first is the $y$-coefficient of the second (and vice-versa), what is the second simplifying step?
Answer
Subtract one equation from the other to obtain a simpler equation, usually in the form $x - y = k$.
Question
To solve equations using cross-multiplication, in what standard form must the equations first be expressed?
Answer
$a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
Question
In the method of cross-multiplication, what is the sequence of coefficients and constants used for the arrow arrangement?
Answer
$b, c, a, b$ (the coefficients of $y$, then constants, then coefficients of $x$, then coefficients of $y$ again).
Question
Cross-multiplication rule: When calculating products between terms connected by arrows, which product is subtracted?
Answer
The product of the numbers connected by the upward arrow.
Question
Cross-multiplication rule: When calculating products between terms connected by arrows, which product is calculated first?
Answer
The product of the numbers connected by the downward arrow.
Question
What is the formula for the variable $x$ in the method of cross-multiplication?
Answer
$x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}$
Question
What is the formula for the variable $y$ in the method of cross-multiplication?
Answer
$y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}$
Question
In the cross-multiplication formulas for $x$ and $y$, what is the common denominator?
Answer
$a_1b_2 - a_2b_1$
Question
How should you solve a system where the variables are only in the denominators, such as $\frac{7}{x} + \frac{8}{y} = 2$?
Answer
Use substitutions such as $a = \frac{1}{x}$ and $b = \frac{1}{y}$ to transform them into linear equations.
Question
For complex denominators like $\frac{30}{x-y} + \frac{44}{x+y} = 10$, what substitution is recommended for $x-y$?
Answer
Let $x - y = a$ (or $\frac{1}{x-y} = a$).
Question
What is the first step in solving a word problem using simultaneous equations?
Answer
Assume the two unknown quantities to be the variables $x$ and $y$.
Question
In problems based on numbers, if the sum of two numbers is $12$ and their difference is $2$, what are the resulting equations?
Answer
$x + y = 12$ and $x - y = 2$.
Question
In fraction problems, how is a fraction typically represented algebraically?
Answer
As $\frac{x}{y}$, where $x$ is the numerator and $y$ is the denominator.
Question
In fraction problems, if the numerator is decreased by $1$, what is the new expression for the numerator?
Answer
$x - 1$
Question
In fraction problems, if the denominator is increased by $5$, what is the new expression for the denominator?
Answer
$y + 5$
Question
How is a two-digit number represented algebraically if $x$ is the tens digit and $y$ is the units digit?
Answer
$10x + y$
Question
How is a two-digit number expressed algebraically after its digits are reversed?
Answer
$10y + x$
Question
If the sum of the digits of a two-digit number is $7$, what is the algebraic equation?
Answer
$x + y = 7$
Question
If a person's present age is $x$ years, what will their age be '$n$' years hence (in the future)?
Answer
$x + n$
Question
If a person's present age is $x$ years, what was their age '$n$' years ago?
Answer
$x - n$
Question
In problems involving Cost Price (C.P.), if the C.P. is $x$ and there is a profit of $25\%$, what is the Selling Price (S.P.)?
Answer
$S.P. = x + 25\% \text{ of } x = \frac{5x}{4}$ (or $1.25x$).
Question
In problems involving Cost Price (C.P.), if the C.P. is $y$ and there is a profit of $10\%$, what is the Selling Price (S.P.)?
Answer
$S.P. = y + 10\% \text{ of } y = \frac{11y}{10}$ (or $1.1y$).
Question
In Time and Work problems, if A can finish a piece of work in $x$ days, what is A's work in one day?
Answer
$\frac{1}{x}$
Question
If A and B together can do a piece of work in $15$ days, what equation represents their combined daily work?
Answer
$\frac{1}{x} + \frac{1}{y} = \frac{1}{15}$, where $x$ and $y$ are the days each takes alone.
Question
In problems based on rectangles, if the length is $x$ and the breadth is $y$, what is the area?
Answer
$xy$
Question
In rectangle problems, if the length is reduced by $5$ units and the breadth is increased by $3$ units, what is the expression for the new area?
Answer
$(x - 5)(y + 3)$
Question
If the digits of a two-digit number differ by $3$, what are the two possible algebraic relationships between the tens digit ($x$) and units digit ($y$)?
Answer
$x - y = 3$ or $y - x = 3$.
Question
For equations like $ax + by = c$, what is the term for $c$?
Answer
The constant term.
Question
In taxi fare problems, if there is a fixed charge $x$ and a per kilometre charge $y$, what is the total fare for a distance of $10$ km?
Answer
$x + 10y$
Question
In problems involving mixtures, if $x$ litres of a $90\%$ solution and $y$ litres of a $97\%$ solution result in $21$ litres of a $95\%$ solution, what is the quantity equation?
Answer
$x + y = 21$
Question
In problems involving mixtures, if $x$ litres of a $90\%$ solution and $y$ litres of a $97\%$ solution result in $21$ litres of a $95\%$ solution, what is the concentration equation?
Answer
$90\% \text{ of } x + 97\% \text{ of } y = 95\% \text{ of } 21$.
Question
What is the algebraic representation of 'the sum of the reciprocals of two numbers $x$ and $y$ is $\frac{4}{21}$'?
Answer
$\frac{1}{x} + \frac{1}{y} = \frac{4}{21}$
Question
If the ratio of the present ages of A and B is $5:3$, how can this be written as an equation using variables $x$ and $y$?
Answer
$\frac{x}{y} = \frac{5}{3}$ or $3x - 5y = 0$.
Question
To verify if a pair of values $(x, y)$ is a solution to a system, what must happen when they are substituted into the equations?
Answer
The values must satisfy both equations (the left-hand side must equal the right-hand side in both).
Question
If an equation is presented as $\frac{x+y}{xy} = 2$, how can it be rewritten in a form that allows for substitution?
Answer
$\frac{x}{xy} + \frac{y}{xy} = 2$, which simplifies to $\frac{1}{y} + \frac{1}{x} = 2$.
Question
In two-digit number problems, if reversing the digits increases the number, is the units digit ($y$) or tens digit ($x$) larger?
Answer
The units digit ($y$) is larger ($y > x$).
Question
In age problems, if a father's age is $x$ and the sum of his two children's ages is $y$, what will the sum of the children's ages be in $20$ years?
Answer
$y + 40$ (because each child's age increases by $20$ years).
Question
When solving simultaneous equations with variables in denominators, is it necessary to find the L.C.M. of the denominators?
Answer
No, it is usually simpler to solve without taking the L.C.M. by using substitution or equating coefficients directly.
Question
In a library charge problem where the first three days have a fixed charge ($x$) and subsequent days have a daily charge ($y$), what is the total charge for $7$ days?
Answer
$x + 4y$
Question
For the system $x + y = 3.3$ and $0.6/(3x-2y) = -1$, what is the first step to simplify the second equation?
Answer
Cross-multiply to get $0.6 = -1(3x - 2y)$, which becomes $3x - 2y = -0.6$.
Question
If $10y = 7x - 4$ and $12x + 18y = 1$, what method is easiest if you want to avoid fractions in the initial steps?
Answer
Method of elimination by equating coefficients.
Question
In rectangle area problems, if length is $x$ and breadth is $y$, what equation is formed if area increases by $67$ when length increases by $3$ and breadth increases by $2$?
Answer
$(x + 3)(y + 2) = xy + 67$
Question
In the cross-multiplication arrow diagram, what does the '1' represent in the sequence $\frac{x}{\dots} = \frac{y}{\dots} = \frac{1}{\dots}$?
Answer
The constant terms in the equations when they are on the left-hand side ($= 0$).