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Chapter 2: Problem 46

Solve each equation, and check the solution. $$ -7.2=-4.5 x $$

Short Answer

Expert verified
Solve: \( x = 1.6 \). Solution verified.

Step by step solution

01

Identify the equation and goal

The equation given is \[ -7.2 = -4.5x \]. The goal is to isolate the variable \( x \).
02

Divide both sides by the coefficient of \( x \)

In order to isolate \( x \), divide both sides of the equation by \( -4.5 \):\[ \frac{-7.2}{-4.5} = \frac{-4.5x}{-4.5} \].
03

Simplify the division

Simplify the left-hand side:\[ \frac{-7.2}{-4.5} = 1.6 \]. Therefore, \( x = 1.6 \).
04

Check the solution

Substitute \( x = 1.6 \) back into the original equation to verify:\[ -7.2 = -4.5(1.6) \]. Calculate the right-hand side:\[ -4.5 \times 1.6 = -7.2 \]. The original equation holds true.

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Key Concepts

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

divide both sides
When you encounter a linear equation, one of the common steps is to divide both sides of the equation. This process helps to simplify the equation, making it easier to isolate the variable. For instance, in the equation \[ -7.2 = -4.5x \], our goal is to solve for x. To do this, we need to get rid of the coefficient attached to x, which in this case is -4.5.
  • Divide both sides by -4.5 to keep the equation balanced.
  • After dividing, the equation looks like \[ \frac{-7.2}{-4.5} = \frac{-4.5x}{-4.5} \] This step simplifies the equation, heading us closer to the solution.Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain equality.
isolate the variable
The idea of isolate the variable is crucial in solving linear equations. To isolate means to get the variable, here x, by itself on one side of the equation. From our equation,\[ -7.2 = -4.5x \], after dividing both sides by -4.5, the next step is to simplify the right-hand side of the equation. Notice how \[ \frac{-4.5x}{-4.5} \] cancels out to just x because any number divided by itself equals 1. Therefore, we now have \[ \frac{-7.2}{-4.5} = x \]This means x is now isolated and we can find its value directly. Isolating the variable essentially simplifies the problem to a basic arithmetic operation.
simplify expression
Simplify expression is the final, yet essential, step to solving linear equations. After isolating the variable x, simplify the remaining expression to find the value of the variable. For instance:\[ x = \frac{-7.2}{-4.5} \]requires you to perform the division.
  • Calculate \[ \frac{-7.2}{-4.5} = 1.6 \]
  • Thus, x = 1.6.

    After finding x, always perform a quick check by substituting the variable back into the original equation to ensure your solution is correct. Here, substitute x = 1.6 back:\[ -7.2 = -4.5(1.6) \]and calculate the right-hand side:\[ -4.5 \times 1.6 = -7.2 \]Since the left-hand side equals the right-hand side, your solution is verified. Simplifying expressions brings us to the final numeric value of the variable, ensuring the equation balances out perfectly.

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