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Chapter 5: Problem 16

Solve the equation. \(-7.9 x+2.9 x=8.6\)

Short Answer

Expert verified
The solution is \(x = -1.72\).

Step by step solution

01

Combine Like Terms

The equation given is \(-7.9x + 2.9x = 8.6\). We need to combine the terms with \(x\): \(-7.9 + 2.9\) is \(-5.0\). So the equation simplifies to \(-5.0x = 8.6\).
02

Isolate the Variable

Now that we have \(-5.0x = 8.6\), we need to solve for \(x\). To do this, divide both sides of the equation by \(-5.0\) to isolate \(x\). Hence:\[x = \frac{8.6}{-5.0}\].
03

Calculate the Division

Perform the division \(8.6 \, \div \, -5.0\). This results in \(x = -1.72\). Thus, the solution is \(x = -1.72\).

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Key Concepts

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

Combining Like Terms
In solving linear equations, combining like terms is often one of the first steps. Like terms have the same variable raised to the same power. In the example equation \(-7.9x + 2.9x = 8.6\), both terms on the left have the variable \(x\). Here, you are essentially combining the coefficients of \(x\), which are \(-7.9\) and \(2.9\).
  • Start by simply adding the coefficients: \(-7.9 + 2.9\).
  • This simplifies to \(-5.0\), meaning the equation now reads \(-5.0x = 8.6\).
Combining like terms simplifies equations, making it easier to isolate the variable in the next steps.
Understanding how to recognize and combine like terms is crucial. It streamlines equations and reduces complexity, prepping the path for further simplifications.
Isolating the Variable
After combining like terms, the next step in solving a linear equation is isolating the variable. The goal is to get the variable, in this case, \(x\), by itself on one side of the equation. With our simplified equation \(-5.0x = 8.6\), we need to manipulate it to solve for \(x\).
  • Here, \(x\) is multiplied by \(-5.0\). To isolate \(x\), we should perform the inverse operation, which is division.
  • By dividing both sides of the equation by \(-5.0\), you effectively cancel out the coefficient attached to \(x\).
  • This leaves us with the expression \[x = \frac{8.6}{-5.0}\].
This step is important as it sets up the equation in a form where \(x\) is now directly solvable. Isolating the variable transforms the equation into one that is easy to handle for the final calculation.
Division in Equations
After isolating the variable in an equation, division often becomes the next focus. We arrived at \[x = \frac{8.6}{-5.0}\]. This division step is crucial as it calculates the precise value of the variable.
  • When dividing, consider both the numerator (8.6) and the denominator (-5.0).
  • Simply perform the division to obtain \(x = -1.72\).
  • Be careful with the signs during division. A positive divided by a negative results in a negative.
This accurate calculation concludes the solving process. By breaking down each operation into manageable steps, division answers the equation and verifies our previous steps.

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Most popular questions from this chapter

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