Solved on Jan 20, 2024

Identify the property of real numbers in the equation: $-5[(-7y)+6y^2]=35y+(-30y^2)$ .

STEP 1

Assumptions
1. We are working within the set of real numbers.
2. We need to identify the property of real numbers that is illustrated by the given equation.

STEP 2

The given equation is: $-5\left[(-7 y)+6 y^{2}\right]=35 y+\left(-30 y^{2}\right)$

STEP 3

We will first expand the left-hand side of the equation using the distributive property of multiplication over addition, which states that for any real numbers $a$ , $b$ , and $c$ , we have $a(b + c) = ab + ac$ .

STEP 4

Apply the distributive property to the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = -5 \cdot (-7 y) + (-5) \cdot 6 y^{2}$

STEP 5

Now, perform the multiplication for each term on the left-hand side.
$-5 \cdot (-7 y) = 35 y$ $(-5) \cdot 6 y^{2} = -30 y^{2}$

STEP 6

Combine the results of the multiplication to rewrite the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = 35 y - 30 y^{2}$

STEP 7

Now, compare the left-hand side after applying the distributive property with the right-hand side of the original equation.
$35 y - 30 y^{2} = 35 y + (-30 y^{2})$

STEP 8

Notice that the left-hand side and the right-hand side of the equation are identical, confirming that the distributive property was used.

STEP 9

Conclude that the property of real numbers illustrated in the given equation is the distributive property.
The property of real numbers illustrated in the equation is the distributive property.

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Identify the property of real numbers in the equation: $-5[(-7y)+6y^2]=35y+(-30y^2)$ .

STEP 1

Assumptions
1. We are working within the set of real numbers.
2. We need to identify the property of real numbers that is illustrated by the given equation.

STEP 2

The given equation is: $-5\left[(-7 y)+6 y^{2}\right]=35 y+\left(-30 y^{2}\right)$

STEP 3

We will first expand the left-hand side of the equation using the distributive property of multiplication over addition, which states that for any real numbers $a$ , $b$ , and $c$ , we have $a(b + c) = ab + ac$ .

STEP 4

Apply the distributive property to the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = -5 \cdot (-7 y) + (-5) \cdot 6 y^{2}$

STEP 5

Now, perform the multiplication for each term on the left-hand side.
$-5 \cdot (-7 y) = 35 y$ $(-5) \cdot 6 y^{2} = -30 y^{2}$

STEP 6

Combine the results of the multiplication to rewrite the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = 35 y - 30 y^{2}$

STEP 7

Now, compare the left-hand side after applying the distributive property with the right-hand side of the original equation.
$35 y - 30 y^{2} = 35 y + (-30 y^{2})$

STEP 8

Notice that the left-hand side and the right-hand side of the equation are identical, confirming that the distributive property was used.

STEP 9

Conclude that the property of real numbers illustrated in the given equation is the distributive property.
The property of real numbers illustrated in the equation is the distributive property.

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Identify the property of real numbers in the equation: −5[(−7y)+6y2]=35y+(−30y2)-5[(-7y)+6y^2]=35y+(-30y^2)−5[(−7y)+6y2]=35y+(−30y2).

STEP 1

Assumptions1. We are working within the set of real numbers.2. We need to identify the property of real numbers that is illustrated by the given equation.

STEP 2

The given equation is: −5[(−7y)+6y2]=35y+(−30y2)-5\left[(-7 y)+6 y^{2}\right]=35 y+\left(-30 y^{2}\right)−5[(−7y)+6y2]=35y+(−30y2)

STEP 3

We will first expand the left-hand side of the equation using the distributive property of multiplication over addition, which states that for any real numbers aaa, bbb, and ccc, we have a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac.

STEP 4

Apply the distributive property to the left-hand side of the equation.−5[(−7y)+6y2]=−5⋅(−7y)+(−5)⋅6y2-5\left[(-7 y)+6 y^{2}\right] = -5 \cdot (-7 y) + (-5) \cdot 6 y^{2}−5[(−7y)+6y2]=−5⋅(−7y)+(−5)⋅6y2

STEP 5

Now, perform the multiplication for each term on the left-hand side.−5⋅(−7y)=35y-5 \cdot (-7 y) = 35 y−5⋅(−7y)=35y (−5)⋅6y2=−30y2(-5) \cdot 6 y^{2} = -30 y^{2}(−5)⋅6y2=−30y2

STEP 6

Combine the results of the multiplication to rewrite the left-hand side of the equation.−5[(−7y)+6y2]=35y−30y2-5\left[(-7 y)+6 y^{2}\right] = 35 y - 30 y^{2}−5[(−7y)+6y2]=35y−30y2

STEP 7

Now, compare the left-hand side after applying the distributive property with the right-hand side of the original equation.35y−30y2=35y+(−30y2)35 y - 30 y^{2} = 35 y + (-30 y^{2})35y−30y2=35y+(−30y2)

STEP 8

Notice that the left-hand side and the right-hand side of the equation are identical, confirming that the distributive property was used.

STEP 9

Conclude that the property of real numbers illustrated in the given equation is the distributive property.The property of real numbers illustrated in the equation is the distributive property.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Identify the property of real numbers in the equation: $-5[(-7y)+6y^2]=35y+(-30y^2)$ .

Assumptions
1. We are working within the set of real numbers.
2. We need to identify the property of real numbers that is illustrated by the given equation.

The given equation is: $-5\left[(-7 y)+6 y^{2}\right]=35 y+\left(-30 y^{2}\right)$

We will first expand the left-hand side of the equation using the distributive property of multiplication over addition, which states that for any real numbers $a$ , $b$ , and $c$ , we have $a(b + c) = ab + ac$ .

Apply the distributive property to the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = -5 \cdot (-7 y) + (-5) \cdot 6 y^{2}$

Now, perform the multiplication for each term on the left-hand side.
$-5 \cdot (-7 y) = 35 y$ $(-5) \cdot 6 y^{2} = -30 y^{2}$

Combine the results of the multiplication to rewrite the left-hand side of the equation.
$-5\left[(-7 y)+6 y^{2}\right] = 35 y - 30 y^{2}$

Now, compare the left-hand side after applying the distributive property with the right-hand side of the original equation.
$35 y - 30 y^{2} = 35 y + (-30 y^{2})$

Conclude that the property of real numbers illustrated in the given equation is the distributive property.
The property of real numbers illustrated in the equation is the distributive property.