When you look at an algebraic expression like $6p + 7q - 5q - 10$, it's easy to feel intimidated by the mix of letters and numbers. Simplifying this expression means combining terms that share the exact same variable and leaving constant numbers where they belong. The final simplified form of $6p + 7q - 5q - 10$ is $6p + 2q - 10$, because $7q$ and $-5q$ combine into $2q$, while $6p$ and $-10$ have no like terms to merge with.
Algebraic reduction forms the bedrock of solving multi-step equations in high school math and standardized testing. Mastering this foundational skill prevents costly sign errors and builds the confidence needed for advanced math. Let's break down the exact mechanics behind combining terms, handling signs, and avoiding common algebraic traps.
What Are Like Terms in Algebra?
Before you can simplify any multi-variable expression, you have to understand what makes two terms "like" terms. Like terms are individual parts of an algebraic expression that contain the exact same variables raised to the exact same powers.
Think of variables as categories of physical items. If you have 7 apples ($7q$) and you give away 5 apples ($5q$), you're left with 2 apples ($2q$). However, you can't combine apples with oranges ($6p$) or loose dollar bills ($10$).
- Variable terms: Terms containing letters like $p$, $q$, $x$, or $y$. Only terms with identical variable letters can be added or subtracted.
- Constant terms: Plain numbers without any attached letters (like $-10$). Constants can always be combined with other constants.
- Unlike terms: Terms with different variables (such as $6p$ and $7q$) or terms with different exponents (such as $q$ and $q^2$). These must remain separate.
According to guidelines from the National Council of Teachers of Mathematics, recognizing structural patterns in expressions is vital before applying operational rules. When examining $6p + 7q - 5q - 10$, you are looking at four distinct terms separated by addition and subtraction signs.
Step-by-Step Breakdown of $6p + 7q - 5q - 10$
Let's walk through the exact process of reducing our target expression cleanly and systematically.
Step 1: Identify and Group Like Terms
Scan the expression from left to right and tag every term by its variable family.
- The expression is: $6p + 7q - 5q - 10$
- The $p$-terms: $6p$ (only one term exists)
- The $q$-terms: $+7q$ and $-5q$ (note that the sign belongs to the term that follows it)
- The constants: $-10$ (only one constant exists)
Step 2: Combine the Coefficients
Rearrange the expression so that like terms sit next to each other using the commutative property: $$6p + (7q - 5q) - 10$$
Now, perform the arithmetic on the coefficients of the matching variables. For the $q$-terms, calculate $7 - 5$, which gives $2$. Attach the variable $q$ back to the result to get $2q$.
Step 3: Write the Final Simplified Expression
Assemble all the pieces that remain. Since $6p$ and $-10$ cannot combine with anything else, they stay as they are: $$6p + 2q - 10$$
This is the fully reduced form. You cannot simplify it any further because $p$ and $q$ represent different unknown values, and $-10$ has no variable component.
Common Mistakes When Simplifying Expressions
Even experienced students make avoidable errors when working with negative signs and multi-variable strings. Keeping these pitfalls in mind will save you points on exams.
- Dropping the negative sign: The subtraction sign in front of $5q$ belongs to the $5$. Treating it as a positive $5q$ would lead to $7q + 5q = 12q$, which is incorrect. Always treat the operation sign as the sign of the term immediately following it.
- Attempting to combine unlike terms: Some students mistakenly try to add $6p$ and $2q$ into $8pq$ or $8p+q$. Remember that addition and subtraction require identical variable parts; you cannot merge $p$ and $q$ terms together.
- Forgetting constants: When an expression ends with a constant like $-10$, it's easy to accidentally leave it out of the final line after doing the variable math. Every original term must be accounted for in the final answer.
For a deeper look into standard algebraic conventions and notation rules, check out the Khan Academy Algebra Foundation.
Frequently Asked Questions
Can $6p + 2q - 10$ be factored further?
No. Because there is no greatest common factor (GCF) that divides evenly into $6$, $2$, and $10$ simultaneously (since $2$ doesn't divide evenly into $7$ or $10$, wait, $2$ divides into $6$, $2$, and $10$, but $p$ and $q$ don't share a common variable factor), the expression cannot be factored using integer coefficients. It is in its simplest polynomial form.
What happens if there is an exponent, like $q^2$?
If the expression were $6p + 7q^2 - 5q - 10$, you still couldn't combine $7q^2$ and $-5q$. Exponents must match precisely just like variables do. $7q^2$ and $5q$ are considered unlike terms.
Does the order of terms in the final answer matter?
Generally, standard mathematical convention lists variables alphabetically ($p$ before $q$) and puts constant numbers at the very end. Writing $6p + 2q - 10$ is standard, though $2q + 6p - 10$ is mathematically equivalent due to the commutative property.
