Algebra: Translate Words to Expressions & Equations

Given $x$ represents one number and $y$ represents another, which expression accurately represents 'twice a number added to three times another is 35'?

If $x$ represents the number of nickels, then $0.05x$ accurately represents the total value of those nickels in dollars.

Express mathematically: 'The value of some toonies in dollars'. Represent the number of toonies as $t$.

If $m$ represents the amount of money invested, then the interest earned at a rate of 5% is represented by ______.

Match each algebraic expression with its verbal description:

Given the system of equations:

$x + y = 377$ $x - y = 107$

What method provides the most direct approach for solving for $x$ and $y$ while minimizing algebraic manipulation?

If Judy's present age is $j$ and Katie's present age is $k$, then the equation $j + k + 10 = 3j$ accurately represents the statement: 'Five years from now the sum of their ages will be three times Judy's present age'.

Raymond pays $2.50 for admission using 13 coins consisting of dimes and quarters only. If $d$ represents the number of dimes, establish an equation that relates $d$ to the number of quarters, $q$.

At a concert, all 450 seats are sold. Student tickets cost $10 and adult tickets cost $16. If $x$ represents the number of student tickets and $y$ represents the number of adult tickets, then the equation representing the total revenue is 10x + 16y = ______.

Match each variable definition to its corresponding representation in the investment scenario:

Scenario: A business invests $180,000, with some at 18% and the rest at 12%.

A certain alloy contains 9% silver, and another contains 14% silver. Determine the system of equations that accurately models the combination of these alloys to produce 10 kg of a 12% silver alloy, where $n$ represents the kilograms of the 9% alloy and $f$ represents the kilograms of the 14% alloy.

If the Anderson family travels a total of 880 km, partly by car at 80 km/h and partly by ferry at 16 km/h, and if $c$ represents the number of hours traveled by car and $f$ represents the number of hours traveled by ferry, then the equation $80c + 16f = 880$ accurately represents the total distance traveled.

The Anderson family traveled 880 km from St. John's to Halifax, partly by car and partly by ferry. If the total trip took 27 hours, with $c$ representing hours by car and $f$ representing hours by ferry, establish an equation relating $c$ and $f$ to the total time.

Given that 'the sum of two numbers' is represented as $x + y$, 'the difference of two numbers' is represented as $x - y$, and 'twice a number' is represented as $2x$, then 'one number is 25% of another number' can be represented algebraically as x = ______.

Match the scenario with the appropriate system of equations:

Given the equation $x = 0.25y$, derived from the statement 'One number is 25% of another number', what transformation of this equation accurately expresses $y$ in terms of $x$?

If $x$ and $y$ represent two numbers that 'differ by 8', then the only correct algebraic representation of this statement is $x - y = 8$.

In a problem involving mixtures of alloys with different silver percentages, if $n$ represents the quantity (in kg) of a 9% silver alloy, what does the expression $0.09n$ represent?

If the Anderson family travels partly by car and partly by ferry, and $c$ represents the hours traveled by car and $f$ represents the hours traveled by ferry, and the equation $c + f = 27$ models the total travel time, then expressing $f$ in terms of $c$ gives us f = ______.

Match the variable designation with its meaning within the context of the word problems:

Given $m$ represents the amount of money invested, which of the following expressions accurately represents 5% of the money invested?

If the sum of two numbers is 12, and one of those numbers is 25% of another number, then it is impossible for both numbers to be positive integers.

In a geometric context, let $x$ and $y$ be the length and width of a rectangle, respectively. Express mathematically the constraint, 'The length exceeds the width by 8'.

The sum of Judy's and Katie's ages is 41. If Judy's age is represented by $j$ and Katie's age equals $k$, this constraint can be expressed mathematically as: $j$ + ______ = 41.

Match the algebraic term with its described relation:

If $x$ represents student tickets and $y$ represents adult tickets for a concert and the equation $x + y = 450$ is given, what does the value '450' primarily represent in the context of the concert's variables?

In the context of investment problems, if $x is invested at 18% and $y is invested at 12%, then the total return on investment translates directly to $x + y$.

In a mixture problem, a metallurgist combines alloy 'A' and alloy 'B' to create a new compound. If $a$ denotes the mass of alloy A and $b$ represents alloy B, what general condition would be necessary to describe an overall additive property for the combination?

For a journey partitioned between travel by car and by ferry, the relationship for calculating net distance integrates component speed and time parameters. Let $d_c$represent the distance done with a car and $d_f$ represent where the ferry was used, such that $d_c$ + $d_f$ = 880 km, with: car hours represented by $c$ and ferry hours represented by $f$ with total time equaling 27. Derive a direct expression associating individual values and let the respective values equal $D_c$ for the total done by car and $D_f$ done by ferry such that if C represents known hours by car where C can vary, derive general relation combining total distance in terms of an unknown f: 80*(27-f)+16*f=______

For a particular problem or set of relations, establish accurate matches for the variables based on the descriptions:

The number of nickels in your collection is represented by $n$. Which mathematical relationship accurately quantifies the total dollar value if: quarter = 0.25 dollars, toonies = 2, dimes equal only 5 cents and value to quantify total value as function of nickels.

If $x$ defines student tickets, and $y$ represents adult counterparts and their sum should equate solely with available seats ($x + y = Constants$), any combination of parameters involving seat totals mandates ticket amount to stay fixed.

Assume the number of seats is fixed when considering the concert revenue. Also suppose 10 corresponds with each seat if the set corresponds solely with graduate tickets and assume adult equals to variable $A$, what does the expression $10 + A define?

For a problem using quantities of different material, if A mass is y, its counterpart, with total is M, can defined in term y via following relation ______, where constant M relates amounts with varying volume composition.

For different situations involving mathematical relation involving a weighted expression: