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Chapter 2: Problem 25

Solve and check. Label any contradictions or identities. $$32-7 x=11$$

Short Answer

Expert verified
x = 3; The solution was verified and it has no contradictions or identities.

Step by step solution

01

Simplify the Equation by Moving the Constant

Start by isolating the variable term. Subtract 32 from both sides to move the constant term:o32 - 7x - 32 = 11 - 32o-7x = -21
02

Solve for the Variable

Next, solve for the variable by dividing both sides of the equation by -7:o\(-7x/-7 = -21/-7\)o\(x = 3\)
03

Check the Solution

To ensure the solution is correct, substitute the value of x back into the original equation and verify:o\(32 - 7(3) = 11\)o\(32 - 21 = 11\)o\(11 = 11\)Since both sides are equal, the solution is verified.
04

Determine if there are any Contradictions or Identities

The equation has been solved for a specific value of x, and the solution was verified successfully. Therefore, the equation is neither a contradiction nor an identity.

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Key Concepts

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

isolating the variable
The first crucial step in solving a linear equation is isolating the variable. This means we need to get the variable by itself on one side of the equation. Let's look at our example, where our equation is initially:

32 - 7x = 11
To isolate the variable term, we need to remove the constant on the same side as the variable. We can do this by performing an operation that makes the constant zero. In our case, we subtract 32 from both sides of the equation:

32 - 7x - 32 = 11 - 32
This simplifies our equation to:

-7x = -21
Now, the variable term (-7x) is isolated on the left side. This prepares us for the next step, which is solving for the variable.
checking the solution
After finding a solution for the variable in a linear equation, it's important to check whether our solution is correct. We do this by substituting our solution back into the original equation. In our example, after solving, we found that:

x = 3
To check if this solution is correct, we'll substitute 3 for x in the original equation:

32 - 7(3) = 11
32 - 21 = 11
11 = 11
Since both sides of the equation are equal after substituting, our solution x = 3 is verified as correct. This confirms that our steps and calculations were accurate.
contradictions and identities
When solving linear equations, sometimes the process reveals certain special cases likes contradictions and identities.
  • **Contradictions:** These occur if, after simplifying, you end up with a statement that is always false, such as 0 = 5. This means that there is no solution to the equation.
  • **Identities:** These occur if you end up with a statement that is always true, such as 0 = 0, indicating that any number can be a solution. This means there are infinitely many solutions.
In our example, the final simplified equation was:

-7x = -21
And we solved to find:

x = 3
The solution worked perfectly when checked, so this equation is neither a contradiction nor an identity. It has exactly one solution.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. Abriana rented a compact car for a business trip. At the time of the rental, she was given the option of prepaying for an entire tank of gasoline at \(\$ 3.099\) per gallon, or waiting until her return and paying \(\$ 6.34\) per gallon for enough gasoline to fill the tank. If the tank holds 14 gal, how many gallons can she use and still save money by choosing the second option?

Are the inequalities \(t>-7\) and \(7<-t\) equivalent? Why or why not?

Translate to an inequality. The temperature is at most \(-3^{\circ} \mathrm{C}\)

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