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Chapter 4: Problem 65

Perform the operation. Subtract \((8 x+2 y)\) from \((-3 x-7 y)\)

Short Answer

Expert verified
The result of subtracting \((8 x+2 y)\) from \((-3 x-7 y)\) is \(-11x - 9y\).

Step by step solution

01

Distribute the Negative Sign

Start by distributing the negative sign through the parentheses to each term of \((8 x+2 y)\). The distributed negative sign changes the signs of each of the terms. Hence, \((-3 x -7 y)-(8 x+2 y)\) becomes \((-3 x -7 y)+ (-1)*(8 x+2 y)\) which simplifies to \((-3 x -7 y)- 8 x-2 y\).
02

Group Like Terms

Next, group the 'x' terms and the 'y' terms together: \(-3 x - 8 x - 7 y - 2 y\).
03

Combine Like Terms

Combine the 'x' terms: \(-3 x - 8 x = -11 x\); Then combine the 'y' terms: \(-7 y - 2 y = -9 y\). The expression becomes \(-11 x -9 y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
When working with algebraic expressions, identifying and working with like terms is essential. Like terms are terms in an algebraic expression that have identical variable parts. This means that they have the same variable(s) raised to the same power(s). For example, in the expression \( -3x - 8x - 7y - 2y \), the terms \(-3x\) and \(-8x\) are like terms because they both contain the variable \(x\). Similarly, \(-7y\) and \(-2y\) are like terms because they share the variable \(y\).
  • To combine like terms, add or subtract the coefficients (the numbers in front of the variables).
  • This helps simplify expressions and solve equations more easily.
Remember, the key to simplifying expressions effectively is to keep track of the variables and their coefficients, ensuring only like terms are combined.
Negative Sign Distribution
Distributing a negative sign across an expression involves applying the negative sign to each term within parentheses. This changes the sign of each term, effectively multiplying each by \(-1\). For instance, if you have the expression \(-(a+b)\), you would distribute the negative sign to get \(-a - b\).

In our original problem, the expression \( (8x + 2y) \) has a negative sign in front when being subtracted from another expression. So, we need to distribute the negative sign to both terms:
  • The positive \(8x\) becomes \(-8x\).
  • The positive \(2y\) becomes \(-2y\).
By distributing the negative sign correctly, we can then proceed to combine terms easily.
Subtraction of Expressions
Subtracting one algebraic expression from another involves carefully applying arithmetic operations. First, write out the entire expression to be subtracted with one important modification: distribute the negative sign as discussed earlier.

For example, to subtract \((8x + 2y)\) from \((-3x - 7y)\):
  • Rewrite it as \((-3x - 7y) - (8x + 2y)\).
  • Apply negative sign distribution to make it \((-3x - 7y - 8x - 2y)\).
Now that both expressions are ready, group and combine like terms:

  • Combine \(-3x\) and \(-8x\) to get \(-11x\).
  • Combine \(-7y\) and \(-2y\) to get \(-9y\).
The final simplified result is \(-11x - 9y\). Subtraction of expressions requires attention to detail, but following these steps makes the process simple and straightforward.

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