The Distributive Property for 7th Grade: Using variables

To solve this problem, we will simplify the expression $8x(5+y)$ using the distributive property.

• Step 1: Recognize the expression as $8x$ multiplied by two terms inside a parenthesis, $5$ and $y$.
• Step 2: Apply the distributive property: $8x(5+y) = 8x \cdot 5 + 8x \cdot y$.
• Step 3: Compute each multiplication:
  - $8x \cdot 5 = 40x$
  - $8x \cdot y = 8xy$
• Step 4: Combine the results from the multiplication steps: $40x + 8xy$.

Thus, the simplified expression of $8x(5+y)$ is $40x + 8xy$.

Comparing with provided answer choices, the correct solution corresponds to choice $1: 40x + 8xy$.

To solve this problem, we'll apply the distributive property and combine like terms.

• Step 1: Apply the distributive property.
  We have the expression $2a + 3b - 6(a - 2b)$. Let's distribute the $-6$ across the terms inside the parentheses:
  $-6(a - 2b) = -6 \cdot a + (-6) \cdot (-2b) = -6a + 12b$.
• Step 2: Combine like terms.
  Now substitute $-6a + 12b$ back into the expression:
  $2a + 3b - 6a + 12b$.
  Combine the like terms, $2a$ and $-6a$, and $3b$ and $12b$:
  $(2a - 6a) + (3b + 12b) = -4a + 15b$.

Therefore, the simplified expression is $-4a + 15b$, which matches choice 2.

To solve this problem, we'll expand and simplify the given expression $4a(a-b) + 3b(a-b)$ by applying the distributive property.

Let's go through the steps:

• Step 1: Apply the distributive property to the first term.
  $4a(a-b) = 4a \cdot a - 4a \cdot b = 4a^2 - 4ab$
• Step 2: Apply the distributive property to the second term.
  $3b(a-b) = 3b \cdot a - 3b \cdot b = 3ab - 3b^2$
• Step 3: Combine the results from Step 1 and Step 2.
  Combine like terms: $4a^2 - 4ab + 3ab - 3b^2 = 4a^2 - ab - 3b^2$

Therefore, the simplified form of the expression is $4a^2 - ab - 3b^2$.

Among the given choices, the correct answer is:

$4a^2-ab-3b^2$

To solve the problem of simplifying the expression $a+3(4a-6)$, follow these detailed steps:

• Step 1: Distribute 3 across the terms in the parentheses.
  This means multiplying 3 with each term inside: $4a$ and $-6$.
• Step 2: Perform the multiplication:
  $3 \times 4a = 12a$
  $3 \times (-6) = -18$
• Step 3: Combine the distributed terms with $a$:
  Start with the given expression, $a + 12a - 18$.
• Step 4: Simplify by combining like terms:
  Add the coefficients of $a$: $a + 12a = 13a$.
• Step 5: Form the final expression:
  The simplified expression is $13a - 18$.

Therefore, the solution to the problem is $13a - 18$, which corresponds to choice 2.

To solve this problem, we'll follow these steps:

• Step 1: Apply the distributive property to expand the expression.
• Step 2: Simplify the expression by combining like terms.

Now, let's work through each step:

Step 1: Apply the distributive property to $6x(-2x + 3y)$.
This gives us:
$6x \cdot (-2x) + 6x \cdot (3y) = -12x^2 + 18xy$.

Step 2: Add this result to $12x^2$:
$-12x^2 + 18xy + 12x^2$.

Combine like terms:
$-12x^2 + 12x^2$ cancels out, leaving $18xy$.

Therefore, the simplified form of the expression is $18xy$.

To solve this problem, we'll follow these steps:

• Step 1: Distribute the first part of the expression $a(a+b)$.
• Step 2: Distribute the second part of the expression $-ab(5a-6b)$.
• Step 3: Combine the expressions obtained from steps 1 and 2.
• Step 4: Combine like terms.

Now, let's work through each step:

Step 1: Distribute $a(a+b)$:
$a(a+b) = a \cdot a + a \cdot b = a^2 + ab$.

Step 2: Distribute $-ab(5a-6b)$:
$-ab(5a-6b) = -ab \cdot 5a + (-ab) \cdot (-6b) = -5a^2b + 6ab^2$.

Step 3: Combine the results obtained from these two distributions:
$(a^2 + ab) + (-5a^2b + 6ab^2)$.

Step 4: Combine like terms:
The expression is already simplified as all terms are unique.
Therefore, the simplified form of the expression is $a^2 + ab - 5a^2b + 6ab^2$.

Therefore, the solution to the problem is $a^2 + ab - 5a^2b + 6ab^2$.

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:

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

$x=2$

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:

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

$x=3$

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:

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

$a=4$

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:

$4,410+9,450x$

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:

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

In order to solve the exercise, we first need to know the total area of the fence.

Let's remember that the area of a rectangle equals length times width.

Let's write the exercise according to the given data:

$7x\times(30x+4)$$$

We'll use the distributive property to solve the exercise. That means we'll multiply 7x by each term in the parentheses:

$(7x\times30x)+(7x\times4)=$

Let's solve each term in the parentheses and we'll get:

$210x^2+28x$

Now to calculate the painting time, we'll use the formula:

$\frac{7m^2}{\frac{1}{2}hr}=14\frac{m^2}{hr}$

The time will be equal to the area divided by the work rate, meaning:

$\frac{210x^2+28x}{14}$

Let's separate the exercise into addition between fractions:

$\frac{210x^2}{14}+\frac{28x}{14}=$

We'll reduce by 14 and get:

$15x^2+2x$

And this is Isaac's work time.