Problem 44 - Prealgebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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:\[-3x = 21 - 6\]\[-3x = 15\]

Step 2: 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 \(-3\):\[x = \frac{15}{-3}\]\[x = -5\]

Step 3: 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.

Key Concepts

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.