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Chapter 2: Problem 28

For exercises 1-72, (a) solve. (b) check. $$ -3 x-18=45 $$

Short Answer

Expert verified
x = -21

Step by step solution

01

Isolate the variable term

To isolate the variable term (-3x), begin by adding 18 to both sides of the equation to get rid of the constant term on the left side. \(-3x - 18 + 18 = 45 + 18\) This simplifies to \(-3x = 63\).
02

Solve for x

Next, solve for (x) by dividing both sides of the equation by (-3): \(\frac{-3x}{-3} = \frac{63}{-3}\) This simplifies to \(x = -21\).
03

Check the solution

To verify the solution, substitute (-21) back into the original equation: \(-3(-21) - 18 = 45\) Calculate as follows: \(63 - 18 = 45\) Since both sides of the equation are equal, \(x = -21\) is the correct solution.

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Key Concepts

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

isolate the variable
The first step in solving linear equations is to isolate the variable. In our example, we start with the equation \(-3x - 18 = 45\). The goal is to get \(x\) by itself on one side of the equation. To do this, we need to eliminate any constants (here, \(-18\)) from the side with the variable. We can achieve this by performing the opposite operation. Since the constant \(-18\) is being subtracted from \(-3x\), we add 18 to both sides of the equation:
  • \[-3x - 18 + 18 = 45 + 18\]
  • This simplifies to \[-3x = 63\]
Now, the variable term \(-3x\) is isolated, and we can move on to the next step.
check the solution
After finding the potential solution, it's essential to check whether it satisfies the original equation. To do this, substitute \(-21\) back into the initial equation and see if both sides remain equal: \(-3(-21) - 18 = 45\). Let's break it down:
  • First, calculate \(-3 \times -21 = 63\)
  • Then, subtract the 18: \[-18\]
  • This results in \[63 - 18 = 45\]
  • Since both sides of the equation equal 45, the solution \(x = -21\) is correct.
By checking the solution, we ensure that our original equation holds true and verify our solution is accurate.
simplify the equation
Simplifying the equation is a critical step which involves combining like terms and reducing any fractions or multipliers. This makes the equation easier to solve. In our example, starting off with:
  • \(-3x - 18 = 45\)
  • After isolating the variable term, we update it to: \[-3x = 63\]
Here, the equation is already simplified and all terms are either combined or isolated appropriately. Simplifying early avoids mistakes and makes our arithmetic clearer.
division in equations
In the final steps when solving for the variable, division often comes into play. We use division to isolate the variable completely. From our simplified equation \( -3x = 63 \), we need to get \( x \) alone:
  • Divide both sides by \( -3 \):
  • \[-3x / -3 = 63 / -3\]
This simplifies to \( x = -21 \). Dividing both sides ensures that the variable \( x \) is by itself, with its coefficient being 1. Remember, whatever operation you perform on one side of the equation, you must do the exact same to the other side to keep it balanced.

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