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Chapter 3: Problem 44

Solve each equation. Check your solution. $$6-3 x=21$$

Short Answer

Expert verified
The solution is \(x = -5\).

Step by step solution

01

Isolate the Variable Term

To begin solving the equation \(6 - 3x = 21\), we need to isolate the term containing the variable \(x\). Subtract 6 from both sides to remove the constant term on the left side of the equation. This gives us:\[-3x = 21 - 6\]\[-3x = 15\]
02

Solve for the Variable

Now, we need to solve for \(x\) by getting rid of the coefficient \(-3\) that is multiplying \(x\). We do this by dividing both sides of the equation by \(-3\):\[x = \frac{15}{-3}\]\[x = -5\]
03

Check the Solution

To verify that \(x = -5\) is the correct solution, substitute it back into the original equation \(6 - 3x = 21\):\[6 - 3(-5) = 21\]Calculate the left side:\[6 + 15 = 21\]\[21 = 21\]The equation holds true, confirming that \(x = -5\) is the correct solution.

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Key Concepts

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

Variable Isolation
Variable isolation is a crucial step in solving linear equations. It involves transforming the equation such that the variable, in this case, \( x \), is alone on one side of the equation. This makes it easier to determine its value. In the equation \(6-3x=21\), the goal is to isolate the \( x \)-term which is the \(-3x\). This means we need to remove other numbers from the side of the equation where the \( x \)-term is situated. In this case, by subtracting \( 6 \) from both sides of the equation, we move toward isolating the \( x \)-term:
  • Original equation: \(6 - 3x = 21\)
  • Subtract \(6\) from both sides: \(-3x = 15\)
This step ensures that \( x \) does not have any additional constants on its side.
Coefficients
Coefficients are numbers that are multiplied by the variables in an equation. In our example, the equation \(-3x = 15\) has \(-3\) as the coefficient of \( x \). Understanding how to manipulate these numbers is key to solving for the variable. To eliminate the coefficient and solve for \( x \), we perform the opposite operation that the coefficient is involved in. Here, \(-3\) is multiplied by \( x \), hence we need to divide by \(-3\) to remove it:
  • Divide both sides by \(-3\): \(x = \frac{15}{-3}\)
  • Simplify the division to get \( x = -5 \)
By doing this, \( x \) is completely isolated, and we find its value.
Solution Checking
Solution checking is the final step to confirm if our solved value for \( x \) is correct. It involves substituting the value back into the original equation and verifying that it maintains equality. This step ensures accuracy and verifies solution correctness. Substituting \( x = -5 \) back into the original equation \(6 - 3x = 21\), we get:
  • Substitute \( x \) with \(-5\): \(6 - 3(-5) = 21\)
  • Simplify: \(6 + 15 = 21\)
  • Check that the equation holds true: \(21 = 21\)
The left-hand side equals the right-hand side, confirming that \( x = -5 \) is indeed the correct solution.

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Most popular questions from this chapter

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