In this video, we will learn how to read and write algebraic expressions and apply this to real-life situations. An algebraic expression is a mathematical expression that consists of variables, numbers, and operations. The value of this expression can change. We will begin by looking at some basic one-step expressions.
We will let any random number be denoted by the letter 𝑛. If we were asked to write the expression for the number four more than 𝑛, this would be 𝑛 plus four. The number six less than 𝑛 would be 𝑛 minus six. The product of five and 𝑛 would be written as five 𝑛, as the word product means “multiply.” This would be the same as five times 𝑛. The expression for a quarter of 𝑛 could be written as 𝑛 over four or one-quarter multiplied by 𝑛.
When looking at the questions in this video, we will need to use all four of these operations together with brackets and exponents or indices.
Express the following in algebraic form: the square of a number 𝑘 which is then multiplied by 19.
In this question, our variable is the letter 𝑘. We’re asked to square this, which is written as 𝑘 to the power of two. Our exponent of 𝑘 is two. We then need to multiply 𝑘 squared by 19. In its simplified form, this is written 19𝑘 squared. When writing the product of a constant and a variable, we always write the constant first. The square of a number 𝑘 which is then multiplied by 19, in algebraic form, is 19𝑘 squared.
We will now look at a second question where we need to write the algebraic expression.
Write “eleven more than sixteen times the number of women” as an algebraic expression, where 𝑤 is the number of women.
In this question, our variable is 𝑤 as this is the number of women. 16 times the number of women is 𝑤 multiplied by 16. We write this as an algebraic expression as 16𝑤. The constant comes before the variable. Our statement wanted 11 more than this, so we need to add 11. The algebraic expression that is 11 more than 16 times the number of women is 16𝑤 plus 11.
We will now look at some questions in real-world context.
The height of the Empire State Building is 164 meters more than three times the height of the Statue of Liberty. Let ℎ be the height of the Statue of Liberty. Write an expression that represents the height of the Empire State Building in terms of ℎ.
We’re told that the Statue of Liberty is ℎ meters tall. The Empire State Building is 164 meters more than three times the height of the Statue of Liberty. Three times the height would be ℎ multiplied by three. As an algebraic expression, we write this as three ℎ. As the Empire State Building is 164 meters more than this, we need to add 164. The expression that represents the height of the Empire State Building in terms of ℎ is three ℎ plus 164. If we knew the actual height of the Statue of Liberty, we could then substitute this value into the expression three ℎ plus 164 to calculate the height of the Empire State Building.
Mason spent 𝑚 minutes practicing the piano on Monday. On Tuesday, he practiced for 20 minutes more than he did on Monday. On Wednesday, he practiced for 35 minutes less than he did on Tuesday. On Thursday, he practiced for three times as long as he did on Monday. On Friday, he practiced for 25 minutes less than he did on Thursday. Write, in its simplest form, an expression that represents the number of minutes he spent practicing on these five days.
Our first step in this question is to find an expression for the number of minutes that Mason spent practicing on each of the five days. On Monday, he spent 𝑚 minutes practicing. On Tuesday, he practiced for 20 minutes more than on Monday. This is equal to 𝑚 plus 20. On Wednesday, he practiced for 35 minutes less than on Tuesday. So this is equal to 𝑚 plus 20 minus 35. As 20 minus 35 is equal to negative 15, this simplifies to 𝑚 minus 15.
On Thursday, Mason practiced for three times as long as on Monday. This is equal to 𝑚 multiplied by three, which we’ll write as three 𝑚. Finally, on Friday, he practiced for 25 minutes less than Thursday. This is equal to three 𝑚 minus 25. The total time he spent practicing is equal to the sum of these five expressions. This is equal to 𝑚 plus 𝑚 plus 20 plus 𝑚 minus 15 plus three 𝑚 plus three 𝑚 minus 25.
We can then group or collect the like terms. Grouping the 𝑚’s gives us nine 𝑚. 20 minus 15 is equal to five. And subtracting 25 from this gives us negative 20. This means that the total amount of time that Mason spent practicing over the five days is 9𝑚 minus 20. If we knew the exact time that Mason spent practicing on the Monday, the value of 𝑚, we could substitute this into the expression to calculate the total time.
Our final question in this video will involve writing an algebraic expression and then simplifying it.
Emma is 𝑥 years old, and Madison is two years older than her. Sophia is seven times as old as Madison, and Natalie is four years older than Sophia. Write and simplify an expression that represents Natalie’s age.
We are told in the question that Emma is 𝑥 years old. As Madison is two years older than her, her age will be 𝑥 plus two. Sophia is seven times as old as Madison. This means that her age will be seven multiplied by 𝑥 plus two. We can put the 𝑥 plus two in parentheses. We could distribute or expand this by multiplying seven by 𝑥 and then seven by two. This is equal to seven 𝑥 plus 14. Therefore, Sophia’s age is seven 𝑥 plus 14.
Finally, we are told that Natalie’s age is four years older than Sophia. This is equal to seven 𝑥 plus 14 plus four. We can simplify this expression by grouping or collecting like terms. 14 plus four is equal to 18. Therefore, Natalie’s age is seven 𝑥 plus 18. We now have expressions for all four girls. Emma is 𝑥 years old, Madison is 𝑥 plus two years old, Sophia is seven 𝑥 plus 14, and Natalie is seven 𝑥 plus 18 years old.
If we were given the value for 𝑥, we could substitute this into the other expressions to calculate each of the girls’ ages, for example, when 𝑥 equals five. This would mean that Emma was five years old. Five plus two is equal to seven. Therefore, Madison was seven years old. Seven multiplied by five is equal to 35. Adding 14 to this means that Sophia is 49 years old. Following the same process for Natalie means that Natalie would be 53 years old. In some questions we see, we will be given this value for 𝑥 which we need to substitute in.
We will now summarize the key points from this video. An algebraic expression is a mathematical expression that consists of variables, numbers, and operations. For example, two 𝑥 plus seven, 𝑥 minus nine divided by seven, and five 𝑥 squared minus two. As we have seen in the video, they can be used to solve real-life problems in context. Our next step, once we’re happy writing algebraic expressions, would be to evaluate them.