Translating Sentences to Equations: Videos & Practice Problems

Translating verbal phrases into algebraic equations involves recognizing key words that correspond to mathematical operations and symbols. Variables often represent unknown numbers, indicated by words like "number," "quantity," or "value." Operations such as addition, subtraction, multiplication, and division are identified through keywords like "sum," "difference," "times," "divided by," and "of."

One crucial symbol to understand is the equal sign, represented by two parallel lines, which denotes that two expressions are equal. Keywords that translate to the equal sign include "equals," "is," "gives," "results in," "yields," "amounts to," "represents," and the phrase "is the same as." Recognizing these keywords helps in forming algebraic equations from sentences.

For example, the phrase "triple a number is 81" can be translated by identifying "triple" as multiplication by three and "a number" as a variable, commonly denoted as \(x\). The word "is" signals the equal sign. Thus, the algebraic equation becomes:

Here, \$3x\( is an expression representing three times the unknown number, and \)81\( is the value it equals.

Another example is the sentence "The sum of a number \)x\( and 12 is the same as three times the number." The phrase "sum of" indicates addition, so we write \)x + 12\(. The phrase "is the same as" translates to the equal sign, and "three times the number" becomes \)3x$. Putting it all together, the equation is:

By identifying these keywords and their corresponding algebraic symbols, one can confidently convert verbal statements into algebraic equations. This skill is foundational for solving problems involving unknown quantities and understanding relationships between variables.

To represent the scenario where Jason has some amount of money, m, in his wallet, and after spending \$18 on a book, he has \$27 left, we can write an equation that models this situation. Since Jason starts with m dollars and spends \$18, the amount of money remaining is the original amount minus the spent amount. This relationship can be expressed as:

\[m - 18 = 27\]

This equation shows that when \$18 is subtracted from Jason's initial money m, the result is \$27, which is the amount he has left. Solving this equation will help determine the total amount of money Jason had before the purchase. This approach demonstrates how to translate real-world situations into algebraic expressions, a fundamental skill in problem-solving and mathematical reasoning.

To represent the total distance d traveled by a bus moving at a constant speed, we start by considering the bus's speed and travel time. The bus travels at a constant speed of 60 kilometers per hour for t hours. The distance covered during this time can be calculated by multiplying the speed by the time, which gives 60 × t kilometers. After this initial travel, the bus covers an additional 35 kilometers. To find the total distance d, we add this extra distance to the distance traveled during the time t. This relationship can be expressed with the equation:

\[d = 60t + 35\]

This equation effectively models the total distance as a function of time, combining the distance covered at a constant speed with the additional fixed distance. Understanding how to translate real-world scenarios into algebraic expressions is essential for solving word problems involving rates, time, and distance.