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

Another example is the sentence "the sum of a number \)x\( and 12 is the same as three times the number." Here, "sum" indicates addition, so the expression is \)x + 12\(. The phrase "is the same as" translates to the equal sign, and "three times the number" becomes \)3x\(. Combining these, the equation is \)x + 12 = 3x\(. This equation states that the sum of \)x\( and 12 equals three times \)x$.

Understanding how to identify and translate these keywords into algebraic expressions and equations is fundamental in solving word problems. This skill allows for the conversion of real-world situations into mathematical models, facilitating problem-solving and analysis.

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 understanding mathematical relationships.

To represent the total distance d traveled by a bus moving at a constant speed, we start by understanding the relationship between speed, time, and distance. The bus travels at a constant speed of 60 kilometers per hour for t hours. The distance covered during this time is calculated by multiplying the speed by the time, which gives us \$60t$ kilometers.

After traveling for t hours, the bus then covers an additional 35 kilometers. To find the total distance d, we add this extra distance to the distance traveled during the initial time period. This results in the equation:

\[d = 60t + 35\]

This equation effectively models the total distance traveled by the bus, combining both the distance covered at a constant speed and the additional distance traveled afterward. Understanding how to translate real-world scenarios into algebraic expressions like this is essential for solving word problems involving motion and distance.