Question

Simplify the expression. $$(5 x-1)(-3)+6$$

Short answer

The simplified form of the expression is \(-15x + 9\).

Step-by-step solution

Distributing Multiplication Over Addition

Using the distributive law, which states that for all real numbers a, b, and c, \(a(b+c) = ab + ac\), distribute \(-3\) into \(5x - 1\). This results in \(-3 * 5x + (-3)*(-1)\).

Multiply

Perform the multiplication operation by multiplying \(-3\) with \(5x\) to get \(-15x\), and \(-3\) with \(-1\) to get \(3\). This results in \(-15x+3\).

Addition with 6

Add 6 to the expression \(-15x+3\) to get \(-15x+3+6\).

Simplify

Lastly, simplify by combining like terms. Here the like terms are constants so combine 3 and 6 to get 9. So the simplification becomes \(-15x + 9\).

Key concepts

Distributive Property

Understanding the distributive property lays the foundation for simplifying algebraic expressions. When faced with an expression like \( (5x - 1)(-3) + 6 \), it might seem complicated at first, but the distributive property offers a systematic approach.

This property lets you 'distribute' a multiplier over terms that are added or subtracted within parentheses. For instance, \( a(b + c) = ab + ac \) is the classic representation of the distributive property in action. In our example, \( -3 \) is distributed over \( 5x \) and \( -1 \) which simplifies to \( -3 \times 5x + (-3) \times (-1) \). It’s just like sharing cookies equally among friends; you're ensuring that each term within the parentheses gets its fair share of the multiplier.

Combining Like Terms

Once distribution of multiplication is completed, combining like terms is often the next logical step. So what are 'like terms'? Quite simply, these are terms in an algebraic expression that contain the same variables raised to the same power. They're like siblings who share common features.

In the distributed expression \( -15x + 3 + 6 \), the like terms are the constants \( 3 \) and \( 6 \). Their similarity lies in the fact that they don’t have any variables attached to them. To combine them, add their coefficients together to get \( 3 + 6 = 9 \). The variable term \( -15x \) remains unchanged, as it doesn’t have a like term to join with. Consider it the only child in the family. The resulting simplified expression is \( -15x + 9 \).

Multiplication in Algebra

Multiplication in algebra follows similar rules as in basic arithmetic. However, instead of just numbers, we're dealing with variables, constants, and sometimes both. The challenge lies in keeping track of the variables and their coefficients.

When the expression \( -3 \times 5x \) is encountered, think of it as multiplying the number \( -3 \) with the coefficient of \( x \) which is \( 5 \), leading to \( -15 \) and thus, \( -15x \) is formed. Multiplying \( -3 \) by the constant \( -1 \) reverses the sign due to the rule 'a negative times a negative equals a positive', yielding a positive \( 3 \).

These multiplicative maneuvers transform abstract algebraic terms into a neat and simplified answer, showcasing the precise beauty of algebra.