Problem 67 - Algebra 1

Question

Simplify the expression. $$-19 a \div 19$$

Step-by-Step Solution

Answer

-a

Step 1: Recognize the operation

The given operation is a division \(-19a \div 19\). The negative sign applies to the entire term \(19a\). Remember, any number divided by itself gives 1.

Step 2: Apply the division operation

Divide the numerical coefficient \(-19\) by \(19\) which will give us \(-1\). Since the coefficients divide out, we're left with the variable \(a\) in the expression.

Step 3: Write down the result

So the simplified expression is \(-a\).

Key Concepts

Division of Algebraic Terms

When simplifying algebraic expressions, understanding the division of algebraic terms is crucial. Division is one of the basic arithmetic operations and it applies to algebraic expressions just as it does to numbers. The division of an algebraic term by another involves separating the numerical coefficient and the variable part.

Consider the expression \( -19a \div 19 \). To simplify, we divide the numerical coefficient of the algebraic term, which is \( -19 \) by the divisor \( 19 \). Since any number divided by itself is \( 1 \), and any number times \( 1 \) remains unchanged, the numerical coefficient simplifies to \( -1 \).

It's important to remember that variables in the algebraic term act as placeholders and do not change during this division, unless the variable term is also present in the divisor. If the same variable exists in the divisor, then the respective variable powers are subtracted, following the exponent rules. In our example, there's only the single variable \( a \) with no corresponding variable in the divisor, so it remains unaffected.

Negative Coefficients

Negative coefficients in algebra are just like negative numbers in arithmetic; they represent the opposite of the positive version of the number. Dealing with negative coefficients is an essential skill when simplifying algebraic expressions.

In our example \( -19a \div 19 \), the \( -19 \) is a negative coefficient. This negative sign means the term is opposite in sign to its positive counterpart (\( +19a \)). When dividing by \( 19 \), remember that a negative divided by a positive yields a negative result. Hence, \( -19 \div 19 = -1 \).

Keeping track of the sign is important as it can change the entire direction of an inequality or could indicate loss or gain in a real-world context. Always ensure that the sign associated with the numerical coefficient is carried through any arithmetic operation.

Numerical Coefficients

Numerical coefficients are the numbers multiplied by the variables in algebraic terms. They provide great insight into the behavior of algebraic expressions and can significantly affect the simplification process.

Take for instance the numerical coefficient in our example \( -19 \) from the term \( -19a \). This coefficient signifies that the variable \( a \) is scaled by \( -19 \). In simplifying expressions, one must perform operations on these coefficients as per the rules of arithmetic. In this division (\( -19a \div 19 \)), you divide the numerical coefficient by \( 19 \) resulting in \( -1 \) as explained in previous sections.

Numerical coefficients can be whole numbers, fractions, decimals, or negative numbers, and understanding how to work with them is essential for correctly simplifying algebraic expressions and solving equations. In many instances, simplifying the numerical coefficients can make an expression much easier to work with and understand.