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