Question

lessons 2 - 1 and 2 - 2 write and evaluate numerical expressions | write algebraic expressions
quick review
mathematical expressions contain numbers, variables, and operations symbols. expressions with numbers and operations are numerical expressions. to evaluate a numerical expression, use order of operations. expressions that also contain variables are variable expressions. you can write an algebraic expression to represent a situation in which some numbers are not known.
example
evaluate the expression 4²+(7 - 12)+14÷2.
evaluate parentheses.
4²+( - 5)+14÷2
evaluate exponents.
16+( - 5)+14÷2
multiply or divide from left to right.
16+( - 5)+7
add or subtract from left to right.
11 + 7, or 18
practice

1. evaluate the expression.

7²+18-(3×2.5)-3³

1. evaluate the expression.

3/4+7(2.5 + 3.8)+2³

1. daniella and two friends go to see a play. they each pay $15 for a ticket, and each buys a bottle of water at intermission. write an algebraic expression to represent the amount of money the friends spend.

lesson 2 - 3 evaluate algebraic expressions
quick review
a variable expression can be evaluated by substituting a given number into the expression in place of the variable.
example
evaluate the expression 3y - 7b; y = 2; b = - 3
substitute 2 for y and - 3 for b.
3(2)-7(-3)
use order of operations to simplify the expression.
6-( - 21)
6 + 21
27
practice
evaluate each expression for the given values.

1. x² + y² x = 7; y = - 5
2. 3a+2(a + b) a = 10, b = - 4
3. 2/3(a - 9)+1/4(6b + 8) a = 6, b = 4

Explanation:

Step1: Evaluate Practice 1 in first section

• Evaluate parentheses first:
• For the expression $7^{2}+[18-(3\times2.5)] - 3^{3}$, calculate $3\times2.5 = 7.5$. Then $18 - 7.5=10.5$.
• Evaluate exponents: $7^{2}=49$ and $3^{3}=27$.
• The expression becomes $49 + 10.5-27$.
• Add and subtract from left - to - right: $49+10.5 = 59.5$, and $59.5 - 27=32.5$.

Step2: Evaluate Practice 2 in first section

• Evaluate parentheses first: For the expression $\frac{3}{4}+7(2.5 + 3.8)+2^{3}$, calculate $2.5 + 3.8 = 6.3$.
• Then $7\times6.3 = 44.1$ and $2^{3}=8$.
• The expression is $\frac{3}{4}+44.1 + 8$.
• $\frac{3}{4}=0.75$, so $0.75+44.1+8=52.85$.

Step3: Write algebraic expression for Practice 3 in first section

• There are 3 people. Each pays $15$ for a ticket and let the cost of a bottle of water be $w$.
• The total amount of money they spend is $3\times(15 + w)=45 + 3w$.

Step4: Evaluate Practice 1 in second section

• Substitute $x = 7$ and $y=-5$ into $x^{2}+y^{2}$.
• $x^{2}=7^{2}=49$ and $y^{2}=(-5)^{2}=25$.
• Then $x^{2}+y^{2}=49 + 25=74$.

Step5: Evaluate Practice 2 in second section

• Substitute $a = 10$ and $b=-4$ into $3a+2(a + b)$.
• First, simplify the expression inside the parentheses: $a + b=10+( - 4)=6$.
• Then $3a=3\times10 = 30$ and $2(a + b)=2\times6 = 12$.
• The result is $30+12=42$.

Step6: Evaluate Practice 3 in second section

• Substitute $a = 6$ and $b = 4$ into $\frac{2}{3}(a - 9)+\frac{1}{4}(6b + 8)$.
• For $\frac{2}{3}(a - 9)$, $a-9=6 - 9=-3$, and $\frac{2}{3}\times(-3)=-2$.
• For $\frac{1}{4}(6b + 8)$, $6b+8=6\times4 + 8=24 + 8=32$, and $\frac{1}{4}\times32 = 8$.
• The result is $-2+8=6$.

Answer:

1. $32.5$
2. $52.85$
3. $45 + 3w$
4. $74$
5. $42$
6. $6$