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Chapter 2: Problem 183

Solve the following equations with constants on both sides. $$60=-21 x-24$$

Short Answer

Expert verified
x = -4

Step by step solution

01

Add 24 to both sides

Start by isolating the term with the variable. Add 24 to both sides of the equation to get rid of the constant term on the right side.\[60 + 24 = -21x - 24 + 24\]Simplifying this, we get:\[84 = -21x\]
02

Divide both sides by -21

Now, divide both sides of the equation by -21 to solve for x.\[\frac{84}{-21} = \frac{-21x}{-21}\]This simplifies to:\[-4 = x\]

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Key Concepts

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

Equations with Constants on Both Sides
To solve a linear equation with constants on both sides, your goal is to isolate the variable. When you encounter terms and numbers on both sides of the equation, it can seem tricky at first. But by following a series of logical steps, you can make it simpler.

For example, consider this equation: \(60 = -21x - 24\).

Notice that both sides have constants. To solve this, we need to move these constants around so that we can isolate the variable (in this case, x) on one side of the equation. Keeping both sides of the equation balanced while rearranging terms is crucial. In this example, we will add 24 to both sides to start the simplification.
Isolating the Variable
Isolating the variable means getting the variable term alone on one side of the equation. This helps in solving equations efficiently.

Let's take our example equation: \(60 = -21x - 24\).
First, we add 24 to both sides: \[60 + 24 = -21x - 24 + 24\]
This simplifies to: \[84 = -21x\]
What we've essentially done here is moved the constant term \( -24\) from the right side of the equation to the left. By doing this, we have succeeded in isolating the term with x on one side.
This step makes it clear that we are aiming to solve for x by progressively simplifying the equation step-by-step until only the variable remains on one side.
Solving for x
The final step in solving linear equations is to isolate x completely to determine its value. From the previous steps, let鈥檚 consider where we left off: \[84 = -21x\]
Now, to solve for x, we need to get rid of the coefficient -21. We do this by dividing both sides of the equation by -21:
\[\frac{84}{-21} = \frac{-21x}{-21}\]
This simplifies to: \[-4 = x\]
After performing this division, we get \[-4 = x\], meaning the value of x is -4.

To summarize:
  • We started by moving constants so that terms with the variable were on one side.
  • Then, we isolated the variable by adding/subtracting terms from both sides of the equation.
  • Finally, we solved for x by dividing both sides by the variable's coefficient.
By following these logical steps, linear equations with constants on both sides can be solved easily.

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