The Distributive Property for 7th Grade: Using variables

$(9+17x)\times(6+1)=420$

Calculate a X

We begin by solving the addition exercise in the right parenthesis:

$(9+17x)\times7=420$

We then multiply each of the terms inside the parentheses by 7:

$(9\times7)+(17x\times7)=420$

We continue by solving each of the exercises inside of the parentheses:

$63+119x=420$

Following this we rearrange the sections whilst maintaining the appropriate sign:

$119x=420-63$

$119x=357$

Finally we divide the two parts by 119:

$\frac{119}{119}x=\frac{357}{119}$

$x=3$

3

$(a+3a)\times(5+2)=112$

Calculate a a

We begin by solving the two exercises inside of the parentheses:

$4a\times7=112$

We then divide each of the sections by 4:

$\frac{4a\times7}{4}=\frac{112}{4}$

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

$a\times7=28$

Remember that:

$a\times7=a7$

Lastly we divide both sections by 7:

$\frac{a7}{7}=\frac{28}{7}$

$a=4$

4

$(7x+3)\times(10+4)=238$

We begin by solving the addition exercise in the right parenthesis:

$(7x+3)+14=238$

We then multiply each of the terms inside of the parentheses by 14:

$(14\times7x)+(14\times3)=238$

Following this we solve each of the exercises inside of the parentheses:

$98x+42=238$

We move the sections whilst retaining the appropriate sign:

$98x=238-42$

$98x=196$

Finally we divide the two parts by 98:

$\frac{98}{98}x=\frac{196}{98}$

$x=2$

2

Calculate the area of the rectangle below in terms of a and b.

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

$S=w⋅h$

When:

S = area

w = width

h = height

We take data from the sides of the rectangle in the figure.$w=b+8$$h=a+3$

We then substitute the above data into the formula in order to calculate the area of the rectangle:

$S=w⋅h = (b+8)(a+3)$

We use the formula of the extended distributive property:

$(a+b)(c+d)=ac+ad+bc+bd$

We substitute once more and solve the problem as follows:

$S=(b+8)(a+3)=(b)(a)+(b)(3)+(8)(a)+(8)(3)$

$(b)(a)+(b)(3)+(8)(a)+(8)(3)=ab+3b+8a+24$

Therefore, the correct answer is option B: ab+8a+3b+24.

Keep in mind that, since there are only addition operations, the order of the terms in the expression can be changed and, therefore,

$ab+3b+8a+24=ab+8a+3b+24$

ab + 8a + 3b + 24

Express the area of the rectangle below in terms of y and z.

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X height

$S=w⋅h$

Where:

S = area

w = width

h = height

We must first extract the data from the sides of the rectangle shown in the figure.

$w=3y$$h=y+3z$

We then insert the known data into the formula in order to calculate the area of the rectangle:

$S=w⋅h=(y+3z)(3y)$

We use the distributive property formula:

$a\left(b+c\right)=ab+ac$

We substitute all known data and solve as follows:

$S=(y+3z)(3y)=(3y)(y+3z)$

$(3y)(y+3z)=(3y)(y)+(3y)(3z)$

$(3y)(y)+(3y)(3z)=3y^2+9yz$

Keep in mind that because there is a multiplication operation, the order of the terms in the expression can be changed, hence:

$(y+3z)(3y)=(3y)(y+3z)$

Therefore, the correct answer is option D: $3y^2+9yz$

$3y^2+9yz$

A building is 21 meters high, 15 meters long, and 14+30X meters wide.

Express its volume in terms of X.

We use a formula to calculate the volume: height times width times length.

We rewrite the exercise using the existing data:

$21\times(14+30x)\times15=$

We use the distributive property to simplify the parentheses.

We multiply 21 by each of the terms in parentheses:

$(21\times14+21\times30x)\times15=$

We solve the multiplication exercise in parentheses:

$(294+630x)\times15=$

We use the distributive property again.

We multiply 15 by each of the terms in parentheses:

$294\times15+630x\times15=$

We solve each of the exercises in parentheses to find the volume:

$4,410+9,450x$

$4410+9450x$

Given the rectangular area 78 cm².

Find X

We know that the area of a rectangle is equal to its length multiplied by its width.

We begin by writing an equation with the available data.

$78=3\times(x+7)$

We then use the distributive property to solve the equation.

That is, we multiply each of the terms inside of the parentheses by 3:

$78=3\times x+3\times7$

$78=3x+21$

We move 21 to the other side and use the appropriate sign:

$78-21=3x$

$57=3x$

Lastly we divide both sides by 3:

$\frac{57}{3}=\frac{3x}{3}$

$x=19$

$19$

Calculate the area of the rectangle

Let's begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

$S=w⋅h$

Where:

S = area

w = width

h = height

We extract the data from the sides of the rectangle in the figure.

$w=x+5$$h=y+2$

We then substitute the above data into the formula in order to calculate the area of the rectangle:

$S=w⋅h=(x+5)(y+2)$

We use the formula of the extended distributive property:

$(a+b)(c+d)=ac+ad+bc+bd$

We once again substitute and solve the problem as follows:

$S=(x+5)(y+2)=(x)(y)+(x)(2)+(5)(y)+(5)(2)$

$(x)(y)+(x)(2)+(5)(y)+(5)(2)=xy+2x+5y+10$

Therefore, the correct answer is option C: xy+2x+5y+10.

^{$xy+2x+5y+10$}

Look at the rectangle in the figure.

What is its area?

We know that the area of a rectangle is equal to its length multiplied by its width.

We begin by writing an equation with the available data.

$(4x+x^2)\times(3x+8+5x)$

Next we use the distributive property to solve the equation.

$(4x\times3x)+(4x\times8)+(4x\times5x)+(x^2\times3x)+(x^2\times8)+(x^2\times5x)=$

We then solve each of the exercises within the parentheses:

$12x^2+32x+20x^2+3x^3+16x^2+5x^3=$

Finally we add up all the coefficients of X squared and all the coefficients of X cubed and we obtain the following:

$48x^2+8x^3+32x$

$8x^3+28x^2+44x$

Which expressions represent the area of the rectangle in the drawing?

1. $56x$

2. $9(3x^2+5x)$

3. $x(3x+5)+9(3x+5)$

4. $32x+x^2$

5. $3x^2+45$

6. $3x^2+32x+45$

3, 6

Gerard plans to paint a fence 7X meters high and 30X+4 meters long.

Gerardo paints at a rate of 7 m² per half an hour.

Choose the expression that represents the time it takes for Gerardo to paint one side of the fence.

$15x^2+2x$ hours