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Chapter 2: Problem 2

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(-7 x=-49\)

Short Answer

Expert verified
The solution is \(x = 7\).

Step by step solution

01

Isolate the variable

We start by isolating the variable \(x\). Currently, the equation is \(-7x = -49\). To isolate \(x\), we need to divide both sides of the equation by \(-7\). This gives us: \[ x = \frac{-49}{-7} \]
02

Simplify the fraction

Now, simplify the fraction \(\frac{-49}{-7}\). We calculate the division: \[ x = 7 \] because dividing \(-49\) by \(-7\) equals \(7\).
03

Verification

To verify the solution, substitute \(x = 7\) back into the original equation: \(-7(7) = -49\). Calculating the left side: \(-49 = -49\), which is a true statement. Thus, \(x = 7\) is indeed the correct solution.

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Key Concepts

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

Solving Linear Equations
Solving linear equations is a fundamental skill in algebra, which involves finding the value of the variable that makes the equation true. Linear equations are equations of the first degree, which means they have variables raised only to the power of one.

To solve a linear equation, you typically follow these steps:
  • Identify the equation and recognize it's a linear equation (like the one in the exercise: \(-7x = -49\)).
  • Manipulate the equation to get the variable on one side and the constant on the other.
  • Use basic arithmetic operations (addition, subtraction, multiplication, or division) to simplify and solve for the variable.
In our example, the equation \(-7x = -49\) is linear because it involves multiplying a variable (\(x\)) by a constant (-7). The goal here is to isolate \(x\) to determine its value.
Variable Isolation
Variable isolation is the process of rearranging an equation in such a way that the variable we're solving for stands alone on one side. In our example, this means having only \(x\) on one side of the equation.
  • The main tool for isolating the variable is using inverse operations to "undo" what's being done to the variable.
  • In the equation \(-7x = -49\), \(x\) is being multiplied by \(-7\).
  • To isolate \(x\), perform the inverse operation; divide both sides by \(-7\).
This math operation results in:\[ x = \frac{-49}{-7} \]Simplifying gives us \(x = 7\). The division essentially "cancels out" the multiplication by \(-7\), leaving \(x\) by itself. Variable isolation is a key step because it helps us find the exact value of \(x\) or any variable in an equation.
Verifying Solutions
Verifying solutions ensures that the value found for the variable actually satisfies the equation. This is a crucial step because it confirms the correctness of our solution.

Here's how you verify a solution:
  • Substitute the solution back into the original equation.
  • Perform the operations to check if both sides of the equation are equal.
In our equation \(-7x = -49\), we found \(x = 7\). By substituting \(x = 7\) back into the equation, calculate:\[-7(7) = -49\]After computing, both sides equal \(-49\), confirming that our solution is indeed correct.

Verification is important because it gives you confidence in your answer, ensuring that the original problem's requirements are met.

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Most popular questions from this chapter

Solve the solutions have been started for you. A 46 -foot piece of rope is cut into three pieces so that the second piece is three times as long as the first piece, and the third piece is two feet more than seven times the length of the first piece. Find the lengths of the pieces.

To make an international telephone call, you need the code for the country you are calling. The codes for Mali Republic, C么te d'Ivoire, and Niger are three consecutive odd integers whose sum is \(675 .\) Find the code for each country.

In your own words, explain why a solution of a word problem should be checked using the original wording of the problem and not the equation written from the wording.

In a runoff election in Georgia for a seat in the U.S. Senate, incumbent Senator Saxby Chambliss received 315,217 more votes than challenger Jim Martin. If the total number of votes cast was \(2,126,491,\) find the number of votes for each candidate. (Source: New York Times)

The code to unlock a student's combination lock happens to be three consecutive odd integers whose sum is 51 . Find the integers.
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