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Chapter 3: Problem 36

Determine whether \((-5,-9)\) is a solution of \(n-2 m=1\).

Short Answer

Expert verified
No, \((-5,-9)\) is not a solution because it does not satisfy the equation.

Step by step solution

01

- Identify the variables

First, identify the variables in the equation. The equation given is \( n - 2m = 1 \). Here, \( n \) and \( m \) are the variables.
02

- Substitute the values

Next, substitute the coordinates \((-5, -9)\) into the equation. Assign \( n = -5 \) and \( m = -9 \).
03

- Perform the substitution

Replace \( n \) with \( -5 \) and \( m \) with \( -9 \) in the equation: \( -5 - 2(-9) = 1 \).
04

- Simplify the equation

Simplify the equation after performing the substitution: \( -5 + 18 = 1 \).
05

- Check the result

Calculate the left side of the equation: \( -5 + 18 = 13 \). Compare this with the right side: \( 13 eq 1 \). Since \( 13 eq 1 \), the point \((-5,-9)\) does not satisfy the equation.

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Key Concepts

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

Coordinate Substitution
Coordinate substitution is a straightforward technique. It means replacing variables in an equation with given values, often from a coordinate point. In this exercise, we use the point \((-5, -9)\)\, which gives us values for \((n, m)\). By placing \(-5\) for \(n\) and \(-9\) for \(m\), we transform the general equation into something specific we can solve.
Variables Identification
Identifying variables is crucial in solving linear equations. Any letter representing unknown numbers is a variable. Here, \( n \) and \( m \) are our variables. Recognize them before doing any substitution. This ensures you replace the correct elements with the given numbers in the equation.
Equation Simplification
Simplifying the equation involves breaking down complex expressions into simple terms. After substituting values, you simplify the resulting expression.

In this case, substituting \( n = -5 \) and \( m = -9 \) turns the equation into:
\(-5 - 2(-9) = 1\).
By solving inside the parentheses first, we get:
\( -5 + 18 = 1 \).
Solution Verification
To verify a solution, compare both sides of the simplified equation. If they equal each other, the coordinate point is a solution; otherwise, it's not.

Here, evaluating \( -5 + 18 = 13 \), we see \( 13 \) does not equal \( 1 \). Thus, \((-5, -9) \)) is not a solution to \( n - 2m = 1 \). Verifying solutions this way confirms the correctness of your work.

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