Envision Math Accelerated Grade 7 Volume 1 Chapter 5 Generate Equivalent Expressions
They are different because in the first one, we have 2 diagrams and we have to find the right amount on each one and then add these amounts.
On the other hand in the second representation, we have 2 variables that we change and easily multiply by constants.
They are similar because they have the same end results and the same constants and coefficients.
Question. Use Commutative and distributive property for the equivalent expression 3x – 12
Given:
3x − 12
3x − 12
Use distributive property a (b + c) = a×b +a×c
3x − 12 = 3(x − 4)
First equivalent expression is 3(x − 4)
Use commutative property a + b = b + a
3x − 12 = −12 + 3x
The second equivalent expression is −12 + 3x.
Hence, the equivalent expressions are 3(x − 4) and −12 + 3x.
Question. Use commutative and distributive property for the equivalent expression \(-\frac{5}{4}x-\frac{3}{4}\)
Given:
\(-\frac{5}{4} x-\frac{3}{4}\) \(-\frac{5}{4} x-\frac{3}{4}\)Use commutative property a + b=b + a
\(-\frac{3}{4}-\frac{5}{4}\) x
First equivalent expression is \(-\frac{3}{4}-\frac{5}{4}\) x
\(-\frac{5}{4} x-\frac{3}{4}\)Use distributive property a × (b + c) = a × b + a × c
\(\frac{1}{4}\) (−5x−3)
Second equivalent expression is \(\frac{1}{4}\) (−5x−3).
Hence, the equivalent expressions are \(-\frac{3}{4}-\frac{5}{4}\) x and \(\frac{1}{4}\)
Question. Use the distributive property to -5(x-2) equivalent expression.
Given:
−5(x−2)
− 5 (x − 2)
Use the distributive property to write an equivalent expression.
Distributive property a × (b+c) = a × b + a × c
−5(x−2)=−5× x − 5 × (−2)
=−5x + 10
Hence, we use the distributive property.
Equivalent expressions are expressions that have the same end value but are written differently.
Commutative property true only for multiplication and addition
Addition: a + b = b + a
Multiplication: a × b = b × a
Subtraction: 4−2 = 2−4
Division: 4 ÷ 2 = 2 ÷ 4
It can’t go with subtraction and division.
The commutative property is true only for multiplication and addition.
Question. Use the associative property when writing equivalent expressions for 2 reasons.
We can use the associative property when writing equivalent expressions for 2 reasons.
We can use it to get 2 values with the same variable in the bracket so we could add/multiply them.
Example:
\(\frac{1}{2} x+\left(\frac{1}{2} x+4\right) \) Use associative property
\(\left(\frac{1}{2} x+\frac{1}{2} x\right)+4\)x + 4
We can use it to get 2 values with variables in the bracket so we could extract the constant in front of these variables by using the distributive property.
\(\frac{1}{2} x+\left(\frac{1}{2} y+4\right) \) Use associative property
\(\left(\frac{1}{2} x+\frac{1}{2} y\right)+4\) Use distributive property
\(\frac{1}{2}\)(x+y)+4
So we can use it to easily add or extract values.
Question. Write an equivalent expression for the -3 + \(\frac{2}{3}\)\frac{1}{3}[/latex]y expression.
Given:
-3 + \(\frac{2}{3}\)y-4-\(\frac{1}{3}\)y
-3 + \(\frac{2}{3}\)y-4-\(\frac{1}{3}\)y
Use commutative property
− 3 − 4 +\(\frac{2}{3}\)y-\(\frac{1}{3}\)y
− 7 + \(\frac{1}{3}\)y
Hence, the equivalent expression is −7 + \(\frac{1}{3}\)y
Question. Write an expression 3(x – 5) values of the expression for x ’s 1,2 and 3 find out x values.
Given:
3(x − 5)
3x − 15
Put x = 1
3(x−5) = 3(1 − 5)
= 3(−4)
= −12
3x − 15 = 3(1) − 15
= 3−15
= −12
Put x = 2
3(x − 5) = 3(2 − 5)
= 3(−3)
= −9
3x−15 = 3(2) − 15
= 6 − 15
= −9
Put x = 3
3(x−5) = 3(3 − 5)
= 3( − 2)
= −6
3x−15 = 3(3) − 15
= 9 −15
= −6
Given that we can see that they have the same values for some \(\backslash[x \mid]^{\prime} \)s so they are equivalent.
Property that makes them equivalent is distributive property.
The values of the expression for x ’s 1,2 and 3 are −12,−9, and −6. So they are equivalent.
Question. Write an equivalent expression for the 4x + \(\frac{1}{2}\) + 2x – 3 expression.
Given:
4x + \(\frac{1}{2}\)+2x-3
4x + \(\frac{1}{2}\) + 2x-3
Use the commutative property
4x + 2x + \(\frac{1}{2}-3\)
6x + \(\frac{5}{2}\)
Hence, the equivalent expression is 6x+\(\frac{5}{2}\)
Question. Write an equivalent expression for the -3(7 + 5g) expression.
We need to write an equivalent expression for the given expression
−3 (7 + 5g)
Using the distributive property, the expression becomes
−3 (7 + 5g) = −3 × 7 + (−3) × 5g
= −21 − 15g
The equivalent expression is −3(7 + 5g) = − 21 −15g
Question. Write an equivalent expression for the (x + 7) + 3y expression.
We need to write an equivalent expression for the given expression
(x + 7) + 3y
Using the associative property, the expression becomes
(x + 7) + 3y= x + (7 + 3y)
The equivalent expression is (x + 7) + 3y = x + (7 + 3y)
Question. Write an equivalent expression for the \(\frac{2}{9}-\frac{1}{5}\)x expression.
We need to write an equivalent expression for the given expression
\(\frac{2}{9}-\frac{1}{5}\)x
Using the distributive property extracting the common factors out, the expression becomes
\(\frac{2}{9}-\frac{1}{5}\)x = \(\frac{1}{5}\)(5×\(\frac{2}{9}\)-x)
= \(\frac{1}{5}\)(\(\frac{10}{9}\) − x)
= \(\frac{1}{5}\) (1.1111……− x)
= \(\frac{1}{5}\)(\(1 . \overline{1}-x\))
The equivalent expression is \(\frac{2}{9}\) – \(\frac{1}{5}\) x = \(\frac{1}{5}\)(\(1 . \overline{1}-x\))
Question. Find the which expression is equivalent to t + 4 + 3 – 2t
We need to find which expression is equivalent to t + 4 + 3 − 2t
The given expression is t + 4 + 3 − 2t
Simplifying it we get
t + 4 +3−2t = t + 7−2t
= t − 2t + 7
= −t + 7
−t + 7 expression is equivalent to t + 4 + 3 − 2t
Question. The distance in feet that karina swims in a race is represented by 4d – 4, where d is the distance for each lap. Write an expression equivalent to 4d – 4.
Given that, the distance in feet that Karina swims in a race is represented by 4d − 4, where d is the distance for each lap.
We need to write an expression equivalent to 4d − 4
The given expression is 4d − 4
Using the distributive property, and extracting the common terms, we get
4d − 4 = 4 × d − 4 × 1
= 4(d − 1)
The equivalent expression is 4(d−1)
Question. Use the associative property to write an expression equivalent to (w + 9) + 3
We need to use the Associative Property to write an expression equivalent to (w + 9) + 3
The given expression is (w + 9) + 3
Using the associative property, we get
(w + 9) + 3 = w + (9 + 3)
The equivalent expression is (w + 9) + 3 = w + (9 + 3)
Question. Maria said the expression -4n + 3 + 9n – 4 is equivalent to 4n find the error Maria likely made.
Given that, Maria said the expression −4n + 3 + 9n − 4 is equivalent to 4n
We need to find the error Maria likely made.
The given expression is −4n + 3 + 9n − 4a
Solving the expression, we get
−4n + 3 + 9n − 4 = −4n + 9n + 3−4
= 5n − 1
The equivalent expression is 5n − 1. The possible error maria made is wrong addition and subtraction.
Maria added the wrong numbers which make her to do the error.
Question. Write an expression equivalent to x – 3y + 4.
We need to write an expression equivalent to x − 3y + 4.
Using the distributive property, and extracting the common terms, we get
x − 3y + 4 = x−3(y−\(\frac{4}{3}\))
= x−3 (y−1.3333…)
= x − 3(\(y−1 . \overline{3}\))
The equivalent expression is =x−3(\(y−1 . \overline{3}\))
Question. The group chat shows the amount of money that each puts in to rent a car for a trip, four friends are combining their money. Use the commutative property to write two equivalent expressions.
Given that, To rent a car for a trip, four friends are combining their money.
The group chat shows the amount of money that each puts in.
One expression for their total amount of money is 189 plus p plus 224 plus q.
We need to use the Commutative Property to write two equivalent expressions.
The given expression is 189 + p + 224 + q
Using the commutative property to find the equivalent expression, we get
189 + p + 224 + q = (189 + 224) + p + q
= 413 + p + q
Another equivalent expression is
189 + p + 224 + q = p + q +( 189 + 224)
= p + q + 413
The two equivalent expressions are 413 + p + q, p + q + 413
Given that, To rent a car for a trip, four friends are combining their money.
The group chat shows the amount of money that each puts in.
One expression for their total amount of money is 189 plus p plus 224 plus q.
If they need $500 to rent a car, we need to find at least two different pairs of numbers that p and q could be.
The given expression is 189 + p + 224 + q
The amount to rent a car will be $500
Thus, the expression becomes
189 + p + 224 + q = 500
413 + p + q = 500
p + q = 500 − 413
p + q = 87
So we need to find two values which added together to get 87 Therefore
p = 40, q = 47
p = 20, q = 67
The two different pairs of numbers are p = 40, q = 47 and p = 20, q = 67
Question. Find the expressions is equivalent to \(\frac{3}{5}x + 3\).
We need to find which of the given expressions is equivalent to
\(\frac{3}{5}x + 3\)Simplifying the given one by one, we get
\(\frac{2}{5}x \) + 3 \(\frac{1}{5}x \)
= \(\frac{2}{5}x \) + \(\frac{16}{5}x \)
\(\frac{4}{5}x \)− \(\frac{1}{5}x \) + 3 = \(\frac{4x− 1x}{5} \) +3
= \(\frac{3}{5}x \) +3
\(\frac{2}{5}x \) + 3 \(\frac{3}{5}x \) − 1= \(\frac{2}{5}x \) +\(\frac{18}{5}x \) −1
= \(\frac{2x+ 18x}{5} \) −1 = 4x −1
1 + \(\frac{3}{5}\) x + 2 = \(\frac{3}{5}\) x+ 3
1 + \(\frac{x}{5}\) + 2 = \(\frac{x}{5}\) + 3
1 + \(\frac{2}{5}\)x + 3 = \(\frac{2}{5}\)x + 4
The expressions that are equivalent to \(\frac{3}{4}\) x + 3 will be \(\frac{4}{5}\) x- \(\frac{1}{5}\) x + 3 and 1 + \(\frac{3}{5}\) x + 2