Simplify 5(x+3)+3(y+1)
The problem asks to perform the algebraic operation of simplification on the given expression: 5(x+3)+3(y+1). This involves distributing the multiplication over the terms inside the parentheses for both sets of terms and then combining like terms if any, to present the expression in a more concise and simplified manner, without changing its value.
$5 \left(\right. x + 3 \left.\right) + 3 \left(\right. y + 1 \left.\right)$
Answer
Solution:
Step 1
Begin by expanding each term in the expression.
Step 1.1
Use the distributive property to expand: $5(x + 3) + 3(y + 1)$.
Step 1.2
Perform the multiplication of $5$ and $3$: $5x + 15 + 3(y + 1)$.
Step 1.3
Again, apply the distributive property to the second term: $5x + 15 + 3y + 3 \cdot 1$.
Step 1.4
Complete the multiplication of $3$ and $1$: $5x + 15 + 3y + 3$.
Step 2
Combine like terms by adding $15$ and $3$: $5x + 3y + 18$.
Knowledge Notes:
The problem-solving process involves simplifying an algebraic expression using the distributive property and combining like terms. Here are the relevant knowledge points:
Distributive Property: This property states that for any real numbers $a$, $b$, and $c$, the equation $a(b + c) = ab + ac$ holds true. It allows us to multiply a single term by each term inside a parenthesis.
Combining Like Terms: This is a process used in algebra to simplify expressions by adding or subtracting terms that have the same variable raised to the same power. In this case, numerical constants are combined.
Multiplication of Numbers: This is a basic arithmetic operation where a number is added to itself a certain number of times. For example, $5 \cdot 3$ means adding $5$ to itself $3$ times, which equals $15$.
Simplification: The process of reducing an expression to its simplest form. This often involves expanding expressions using the distributive property, combining like terms, and performing arithmetic operations.
In the given problem, the distributive property is applied twice to expand the terms $5(x + 3)$ and $3(y + 1)$, and then simple arithmetic is used to combine the constants $15$ and $3$ to arrive at the simplified expression $5x + 3y + 18$.
