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Chapter 5: Problem 14

Solve the equation. \(0.9 x+4.5-0.5 x=3.5\)

Short Answer

Expert verified
The solution to the equation is \(x = -2.5\).

Step by step solution

01

Combine like terms

First, we need to simplify the equation by combining like terms. Note that we have like terms on the left side of the equation: \(0.9x\) and \(-0.5x\). Combining these gives us:\[0.9x - 0.5x = 0.4x\]Thus, the equation now becomes:\[0.4x + 4.5 = 3.5\]
02

Isolate the variable term

To isolate the variable term \(0.4x\), we need to eliminate the constant \(4.5\) from the left side of the equation. We do this by subtracting \(4.5\) from both sides:\[0.4x + 4.5 - 4.5 = 3.5 - 4.5\]This simplifies to:\[0.4x = -1\]
03

Solve for the variable

To solve for \(x\), we need to divide both sides of the equation by \(0.4\):\[x = \frac{-1}{0.4}\]Simplifying the division, we get:\[x = -2.5\]

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential skill in algebra, as it allows us to simplify equations more effectively. When we refer to "like terms," we're talking about terms that have the same variable raised to the same power. For example, in the equation given: \(0.9x + 4.5 - 0.5x = 3.5\), the terms \(0.9x\) and \(-0.5x\) are like terms because they both contain the variable \(x\).To combine like terms, we simply add or subtract their coefficients. In this case, we subtract \(0.5x\) from \(0.9x\):
  • The coefficients are 0.9 and -0.5.
  • Subtract: \(0.9 - 0.5 = 0.4\).
This results in \(0.4x\), allowing us to rewrite the equation as \(0.4x + 4.5 = 3.5\). This process helps to reduce the number of terms and makes solving for the variable simpler.
Isolating the Variable
Isolating the variable is a crucial step when solving equations. It involves getting the variable term by itself on one side of the equation, so we can determine its value. After combining like terms in our equation, we have \(0.4x + 4.5 = 3.5\). Our goal here is to isolate \(x\).To isolate the variable term, we need to remove the constant term from the left side. We do this by performing the same operation on both sides of the equation to maintain balance:
  • Subtract 4.5 from both sides: \(0.4x + 4.5 - 4.5 = 3.5 - 4.5\).
This results in \(0.4x = -1\). Now the variable \(x\) is isolated with a coefficient of \(0.4\). Remember, maintaining the balance of the equation is important every step of the way.
Simplifying
Simplifying is the process where we make an equation or expression easier to work with by reducing it to its most basic form. After isolating the variable in our problem, we have \(0.4x = -1\). Now, the next task is to find the value of \(x\).To do this, we need to "undo" the multiplication of \(x\) by 0.4. We do this by dividing both sides by 0.4. This process of division can be visualized as breaking down the equation further to reveal the solution:
  • Divide both sides by 0.4: \(x = \frac{-1}{0.4}\).
  • Simplify the fraction: \(x = -2.5\).
This simplification tells us that \(x\) equals -2.5. Each step鈥攃ombining like terms, isolating the variable, and simplifying鈥攂rings us closer to understanding the essence and solving the equation successfully.

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