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Chapter 2: Problem 34

Solve each equation. $$-8 n-17+9 n-4=-41$$

Short Answer

Expert verified
n = -20

Step by step solution

01

Combine like terms

First, combine the terms with the variable n on the left side of the equation. The equation is -8n + 9n - 17 - 4 = -41.Combine -8n and 9n to get n.
02

Simplify constants

Next, combine the constants on the left side of the equation. Combine -17 and -4 to get -21.The simplified equation is now n - 21 = -41.
03

Isolate the variable

To isolate n, add 21 to both sides of the equation: n - 21 + 21 = -41 + 21.This simplifies to n = -20.

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Key Concepts

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

combine like terms
Combining like terms is a crucial step in solving linear equations. It involves merging similar terms to simplify an equation. For instance, in the equation given, $$-8n - 17 + 9n - 4 = -41,$$ you can observe that both \(-8n\) and \(9n\) are like terms because they contain the same variable, \(n\). By combining them together, we sum their coefficients: $$-8 + 9 = 1.$$ Hence, \(-8n + 9n\) becomes simply \(n\). This step helps to simplify complex equations and makes them easier to solve.
isolate the variable
Isolating the variable means solving the equation in terms of just one variable to get its value. It usually involves moving the variable to one side of the equation and constants to the other side. For the equation $$ n - 21 = -41, $$ you need to isolate \(n\) by adding \(21\) to both sides. This way, you eliminate the \(-21\) on the left-hand side. When you perform this operation: $$ n - 21 + 21 = -41 + 21, $$ you end up with $$ n = -20.$$ Now, the variable \(n\) is isolated, giving its value directly.
simplifying constants
Simplifying constants involves adding or subtracting the numbers without variables on both sides of the equation. This process further reduces the equation to a simpler form. In the example, after combining like terms, the equation was $$ n - 21 = -41. $$ To simplify the constants, you perform the addition: $$ -17 + (-4) = -21. $$ This leaves the equation in the simplified form of $$ n - 21 = -41, $$ which then allows you to isolate the variable \(n\) as previously described. Simplifying constants is essential for achieving the simplest form of the equation and making the variable isolation straightforward.

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