Variables and Algebraic Expressions: Solving an equation using the distributive law

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 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 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 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 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 $1: 40x + 8xy$.