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Chapter 2: Problem 14

Solve and check. Label any contradictions or identities. $$5 x-9=41$$

Short Answer

Expert verified
The solution is \(x = 10\).

Step by step solution

01

Isolate the variable term

To isolate the term containing the variable, we need to get rid of the constant term on the left side. Add 9 to both sides of the equation:\(5x - 9 + 9 = 41 + 9\)Simplifying, we get:\(5x = 50\)
02

Solve for the variable

Now that we have isolated the term with the variable, divide both sides of the equation by 5 to solve for \(x\):\(\frac{5x}{5} = \frac{50}{5}\)Simplifying, we get:\(x = 10\)
03

Check the solution

To verify the solution, substitute \(x = 10\) back into the original equation:\(5(10) - 9 = 41\)Simplifying the left-hand side, we get:\(50 - 9 = 41\)\(41 = 41\)Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Isolating the Variable
One of the key steps in solving linear equations is isolating the variable. This means we need to get the variable term by itself on one side of the equation. In our given example, the equation is \(5x - 9 = 41\).To isolate the variable term \(5x\), we must eliminate the constant \(-9\) from the left side. We do this by adding \(9\) to both sides of the equation:\(5x - 9 + 9 = 41 + 9\).Simplifying further, we get \(5x = 50\). By isolating the variable, we make the equation easier to solve.
Integer Division
Once the variable term is isolated, the next step is to get the variable alone. In our example, we have \(5x = 50\).Here, we divide both sides of the equation by \(5\)to solve for \(x\): \(\frac{5x}{5} = \frac{50}{5}\).This simplifies to \(x = 10\). This step involves integer division, which means dividing two whole numbers.
Checking Solutions
An important step in solving equations is to check your solution to ensure it's correct. To do this, substitute your solution back into the original equation. In the provided example, our solution is \(x = 10\).To check, we substitute \(x = 10\)back into the original equation: \(5(10) - 9 = 41\).Simplifying the left-hand side, we get: \(50 - 9 = 41\)and\(41 = 41\).Since both sides of the equation are equal, our solution \(x = 10\)is correct.
Constants in Equations
When solving linear equations, constants are fixed values that do not contain variables. In our example, \(5x - 9 = 41\), \(-9\) and \(41\) are constants. These constants are numbers that we work with during the process of isolating the variable. We often need to move constants from one side of the equation to the other by performing operations like addition or subtraction. This helps us to isolate the variable and eventually solve the equation.

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Most popular questions from this chapter

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