Question Determine if the statement is false or identify the property that justifies it. If cancellation is used, specify the quantity added/multiplied to both sides. $-4x + 14y^6 - z = 8y^6 - z \Leftrightarrow -4x + 6y^6 = 0$

Studdy Solution

STEP 1

Assumptions1. The given statement is-4 x+14 y^{6}-z=8 y^{6}-z \eftrightarrow-4 x+6 y^{6}=0. We need to identify if this statement is false or if it is justified by a mathematical property. If it is justified by a mathematical property, we need to identify which property it is and the quantity that is added to or multiplied by both sides.

STEP 2

Let's first simplify the given statement to see if it is true or false.
The given statement is equivalent to two equations $-4 x+14 y^{6}-z=8 y^{6}-z$ $-4 x+6 y^{6}=0$

STEP 3

Let's subtract $8y^{6}-z$ from both sides of the first equation to see if it simplifies to the second equation.
$(- x+14 y^{6}-z) - (8 y^{6}-z) =0$

STEP 4

implify the left-hand side of the equation.
$-4x +14y^{6} - z -8y^{6} + z = -4x +6y^{6}$

STEP 5

As you can see, the left-hand side of the equation simplifies to $-4x +y^{}$ , which is the left-hand side of the second equation. Therefore, the given statement is true.

STEP 6

Now, let's identify the property that justifies this transformation. In this case, we subtracted the same quantity, $8y^{6}-z$ , from both sides of the equation. This is justified by the Additive Cancellation Property.

STEP 7

The quantity that is added to or multiplied by both sides in this case is $- (y^{6}-z)$ .
So, the answer is Additive Cancellation Property and the quantity is $y^{6}-z$ .

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Question Determine if the statement is false or identify the property that justifies it. If cancellation is used, specify the quantity added/multiplied to both sides. $-4x + 14y^6 - z = 8y^6 - z \Leftrightarrow -4x + 6y^6 = 0$

Studdy Solution

STEP 1

STEP 2

Let's first simplify the given statement to see if it is true or false.
The given statement is equivalent to two equations $-4 x+14 y^{6}-z=8 y^{6}-z$ $-4 x+6 y^{6}=0$

STEP 3

Let's subtract $8y^{6}-z$ from both sides of the first equation to see if it simplifies to the second equation.
$(- x+14 y^{6}-z) - (8 y^{6}-z) =0$

STEP 4

implify the left-hand side of the equation.
$-4x +14y^{6} - z -8y^{6} + z = -4x +6y^{6}$

STEP 5

As you can see, the left-hand side of the equation simplifies to $-4x +y^{}$ , which is the left-hand side of the second equation. Therefore, the given statement is true.

STEP 6

Now, let's identify the property that justifies this transformation. In this case, we subtracted the same quantity, $8y^{6}-z$ , from both sides of the equation. This is justified by the Additive Cancellation Property.

STEP 7

The quantity that is added to or multiplied by both sides in this case is $- (y^{6}-z)$ .
So, the answer is Additive Cancellation Property and the quantity is $y^{6}-z$ .

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Question Determine if the statement is false or identify the property that justifies it. If cancellation is used, specify the quantity added/multiplied to both sides. −4x+14y6−z=8y6−z⇔−4x+6y6=0-4x + 14y^6 - z = 8y^6 - z \Leftrightarrow -4x + 6y^6 = 0−4x+14y6−z=8y6−z⇔−4x+6y6=0

Studdy Solution

STEP 1

STEP 2

Let's first simplify the given statement to see if it is true or false.The given statement is equivalent to two equations−4x+14y6−z=8y6−z-4 x+14 y^{6}-z=8 y^{6}-z−4x+14y6−z=8y6−z−4x+6y6=0-4 x+6 y^{6}=0−4x+6y6=0

STEP 3

Let's subtract 8y6−z8y^{6}-z8y6−z from both sides of the first equation to see if it simplifies to the second equation.(−x+14y6−z)−(8y6−z)=0(- x+14 y^{6}-z) - (8 y^{6}-z) =0(−x+14y6−z)−(8y6−z)=0

STEP 4

implify the left-hand side of the equation.−4x+14y6−z−8y6+z=−4x+6y6-4x +14y^{6} - z -8y^{6} + z = -4x +6y^{6}−4x+14y6−z−8y6+z=−4x+6y6

STEP 5

As you can see, the left-hand side of the equation simplifies to −4x+y-4x +y^{}−4x+y, which is the left-hand side of the second equation. Therefore, the given statement is true.

STEP 6

Now, let's identify the property that justifies this transformation. In this case, we subtracted the same quantity, 8y6−z8y^{6}-z8y6−z, from both sides of the equation. This is justified by the Additive Cancellation Property.

STEP 7

The quantity that is added to or multiplied by both sides in this case is −(y6−z)- (y^{6}-z)−(y6−z).So, the answer is Additive Cancellation Property and the quantity is y6−zy^{6}-zy6−z.

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Question Determine if the statement is false or identify the property that justifies it. If cancellation is used, specify the quantity added/multiplied to both sides. $-4x + 14y^6 - z = 8y^6 - z \Leftrightarrow -4x + 6y^6 = 0$

Let's first simplify the given statement to see if it is true or false.
The given statement is equivalent to two equations $-4 x+14 y^{6}-z=8 y^{6}-z$ $-4 x+6 y^{6}=0$

Let's subtract $8y^{6}-z$ from both sides of the first equation to see if it simplifies to the second equation.
$(- x+14 y^{6}-z) - (8 y^{6}-z) =0$

implify the left-hand side of the equation.
$-4x +14y^{6} - z -8y^{6} + z = -4x +6y^{6}$

As you can see, the left-hand side of the equation simplifies to $-4x +y^{}$ , which is the left-hand side of the second equation. Therefore, the given statement is true.

Now, let's identify the property that justifies this transformation. In this case, we subtracted the same quantity, $8y^{6}-z$ , from both sides of the equation. This is justified by the Additive Cancellation Property.

The quantity that is added to or multiplied by both sides in this case is $- (y^{6}-z)$ .
So, the answer is Additive Cancellation Property and the quantity is $y^{6}-z$ .