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Course: Praxis Core Math > Unit 1

Lesson 4: Algebra
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Praxis Core Math
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Algebra
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Algebraic properties | Lesson

Google Classroom

What properties apply to algebraic expressions and equations?

follow the same set of rules as numerical expressions and equations. The order of operations is preserved, and the laws governing addition and multiplication still apply.
However, with the inclusion of variables comes a new set of challenges. Instead of having a definitive value for an expression, we need to evaluate an algebraic expressions for specific values of the variables, observe how the value of the expression changes as the values of the variables change, and determine statements that are true for all values of the variables.

What skills are tested?

  • Using algebraic laws
  • Evaluating algebraic expressions
  • Using structures in algebra to find values and expressions
  • Understanding how changes in a variable's value affects the value of an expression or another variable

What are some algebraic laws?

Like operations with numbers, operations with variables obey certain laws.
The commutative law tells us we can reorder the terms when performing addition or multiplication. For variables a and b:
  • a+b=b+a
  • a×b=b×a
The associative law tells us that in addition or multiplication, we can associate the terms or factors as we please. For variables a, b, and c:
  • (a+b)+c=a+(b+c)
  • (a×b)×c=a×(b×c)
Both the commutative law and the associative law apply to either addition or multiplication, but not a mixture of the two.
The distributive law deals with the combination of addition and multiplication. When a sum is multiplied by value, the value is distributed to each part of the sum. For variables a, b, and c:
  • a(b+c)=a×b+a×c
The law can also be extended to additional variables and differences.
When verifying whether two algebraic expressions are equal to each other, we can either recall the relevant laws or substitute values for the variables to test whether the expressions are equal. While testing the expressions at specific values does not account for all possible values of the variables, it will help us eliminate incorrect choices.

How do we evaluate algebraic expressions?

To evaluate an algebraic expression:
  1. Replace the variables in the algebraic expression with their respective values.
  2. Perform the operations in the expression in the correct order.
  3. Write the result.

What are structures in algebra?

With the knowledge of algebraic laws, we may recognize structures, or the relationship between algebraic expressions. For example, just as we recognize that 2x is 2 times x, we might recognize that 2(x+y) is equal to 2x+2y.
Recognizing the relationship between algebraic expressions can help us solve for the values of expressions even if we don't know the values of the variables. For example, if we know the value of x+y, then we can find out the value of 2x+2y because the two expressions are related.
When solving questions about structures in algebra:
  1. Identify the relationship between expressions.
  2. If an expression has a known value, write the other expression in terms of the expression with known value.
  3. Substitute the known value for the expression.
  4. Evaluate the expression.

How does changing the value of a variable affect other parts of the equation?

Given an equation, we can maintain the equality as long as we perform the same operation on both sides of the equation.
When we work with equations such as xy=z or xy=z and want to maintain the equality, then:
  • Any changes to one side of the equation must also be applied to the other side of the equation.
  • If one side of the equation remains unchanged, then changes to the other side of the equation must undo each other.
To determine how changing the values of variables affects the value of another variable in the equation:
  1. Determine how the variables are related and write the corresponding equation.
  2. Fill in the known changes to variables.
  3. Calculate the change to the variable in question.

Your turn!

TRY: APPLYING ALGEBRAIC LAWS
Which of the following statements are true for all real numbers x, y, and z ?
Choose all answers that apply:

TRY: EVALUATING ALGEBRAIC EXPRESSION
For x=4 and y=−3, what is the value of 2x+3y ?
Choose 1 answer:

TRY: USING ALGEBRAIC STRUCTURE
If 3x+5y=9, what is the value of 15x+25y ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

TRY: DETERMINING THE EFFECT OF CHANGING A VARIABLE
12XY=Z
In the formula above, if the value of X is doubled and the value of Y is tripled, how will the value of Z change?
Choose 1 answer:

Things to remember

Commutative law:
  • a+b=b+a
  • a×b=b×a
Associative law:
  • (a+b)+c=a+(b+c)
  • (a×b)×c=a×(b×c)
Distributive law:
  • a(b+c)=a×b+a×c
To evaluate an algebraic expression:
  1. Replace the variables in the algebraic expression with their respective values.
  2. Perform the operations in the expression in the correct order.
  3. Write the result.
When solving questions about structures in algebra:
  1. Identify the relationship between expressions.
  2. If an expression has a known value, write the other expression in terms of the expression with known value.
  3. Substitute the known value for the expression.
  4. Evaluate the expression.
When we work with equations such as xy=z or xy=z and want to maintain the equality, then:
  • Any changes to one side of the equation must also be applied to the other side of the equation.
  • If one side of the equation remains unchanged, then changes to the other side of the equation must undo each other.
To determine how changing the values of variables affects the value of another variable in the equation:
  1. Determine how the variables are related and write the corresponding equation.
  2. Fill in the known changes to variables.
  3. Calculate the change to the variable in question.

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  • blobby green style avatar for user musicdunc6
    Posted 5 years ago. Direct link to musicdunc6's post “What happened to the 1/2 ...”
    What happened to the 1/2 in front of x though?
    (11 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user Gianfranco Mellone
      Posted 4 years ago. Direct link to Gianfranco Mellone's post “musicdunc6, For that que...”
      Good Answer
      musicdunc6,

      For that question, I did as the video commentator previously suggested in other sections and plugged in random numbers for X and for Y. I suggest using simple numbers.

      For example, I used 3 for X and 6 for Y and plugged them in.

      Step 1) 1/2XY = Z
      Step 2) 1/2 * 3(6) = Z
      Step 3) 1/2 * 18 = Z
      Step 4) 9 = Z

      If X is doubled and Y is tripled, X = 3 becomes X = 6 and Y = 6 becomes Y = 18. So:

      Step 1) 1/2 * 3(6) = Z now becomes 1/2 * 6(18) = Z
      Step 2) 1/2 * 108 = Z
      Step 3) 54 = Z

      9 X 6 = 54. Therefore, If X is doubled and Y is tripled, Z becomes 6 times as great.
      (10 votes)
  • piceratops seed style avatar for user Julie Schmidt
    Posted 5 years ago. Direct link to Julie Schmidt's post “Im also having difficulty...”
    Im also having difficulty with the 3x + 5y = 9 15x + 25 y

    I was trying to solve for x and y (to check my math) and 45 ( which is the solution), doesn't make sense to me.
    (7 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user Gianfranco Mellone
      Posted 4 years ago. Direct link to Gianfranco Mellone's post “Julie, For that questio...”
      Julie,

      For that question, I did as the video commentator previously suggested in other sections and plugged in random numbers for X and for Y. I suggest using simple numbers.

      For example, I used 3 for X and 0 for Y and plugged them in.

      Step 1) 3x + 5y = 9
      Step 2) 3(3) + 5(0) = 9
      Step 3) 9 + 0 = 9

      Therefore, whichever numbers you chose to use for X and for Y must also be true for the second equation if they worked for the first equation. So:

      Step 1) 15x + 25y = ?
      Step 2) 15(3) + 25(0) = ?
      Step 3) 45 + 0 = 45
      (8 votes)
  • blobby green style avatar for user Shekhar Maheshwari
    Posted 3 years ago. Direct link to Shekhar Maheshwari's post “What grade is this for”
    What grade is this for
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user robsanzone
    Posted 3 years ago. Direct link to robsanzone's post “How come we see this rule...”
    How come we see this rule: "Given an equation, we can maintain the equality as long as we perform the same operation on both sides of the equation."
    But then the next example down says to "double X" but the solution does not show us doubling Z?

    this example:

    If xy=z, how will the value of
    y change if the value of
    x is doubled and the value of
    z remains unchanged?
    The value x is doubled, or multiplied by
    2. The value of
    z remains unchanged, or multiplied by
    1. The product of
    2 and the change to
    y must equal
    1. Since the product of a number and its reciprocal is
    1, the value of
    y must be halved, or multiplied by
    ½. The value of
    y will be ½
    (3 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Doyle Haynes
      Posted a year ago. Direct link to Doyle Haynes's post “Because the question asks...”
      Because the question asks that after x is multiplied by 2 what must be done to y so that the value of z remains the same, y must be multiplied by 1/2 to offset the change to x to this side of the equation.

      The communitive law allows us to take 2*x*(1/2)*y=z and rearrange it to 2*(1/2)*x*y=z and since 2*(1/2)=1 the value of x*y does not change and the value z remains unaffected as well.
      (3 votes)
  • blobby green style avatar for user LucionW
    Posted 2 years ago. Direct link to LucionW's post “I have no questions.”
    I have no questions.
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Myronda Jenkins
    Posted 9 months ago. Direct link to Myronda Jenkins's post “what happened to the 1/2 ...”
    what happened to the 1/2 in that problem?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user abc=123-e=mc2
    Posted a year ago. Direct link to abc=123-e=mc2's post “I don't get the "Calculat...”
    I don't get the "Calculate the change to the variable in
    question" part. Can you explain?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • piceratops seedling style avatar for user mateosg
    Posted 4 months ago. Direct link to mateosg's post “c 3 -5 2 -7 3 d -2 9 -5 1...”
    c 3 -5 2 -7 3
    d -2 9 -5 11 -3
    c − d
    c − (−d)
    −c − d
    −c − (−d)
    hox can i solve this table
    (2 votes)
    Default Khan Academy avatar avatar for user