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
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
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
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
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.