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Chapter 2: Problem 10

Solve and check. Label any contradictions or identities. $$6 z+3=-45$$

Short Answer

Expert verified
The solution is \(z = -8\).

Step by step solution

01

- Isolate the variable term

Subtract 3 from both sides of the equation to isolate the term with the variable.\[ 6z + 3 - 3 = -45 - 3 \]This simplifies to:\[ 6z = -48 \]
02

- Solve for the variable

Divide both sides of the equation by 6 to solve for \(z\).\[ \frac{6z}{6} = \frac{-48}{6} \]This simplifies to:\[ z = -8 \]
03

- Check the solution

Substitute \(z = -8\) back into the original equation to check the solution.Original equation:\[ 6z + 3 = -45 \]Substitute \(z = -8\):\[ 6(-8) + 3 = -45 \]This simplifies to:\[ -48 + 3 = -45 \]\[ -45 = -45 \]The left-hand side equals the right-hand side, confirming that \(z = -8\) is a valid solution.

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Key Concepts

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

Isolate the Variable
When solving linear equations, the first goal is to isolate the variable on one side of the equation. By doing this, we can more easily solve for the variable. In our example, we start with the equation:

\[ 6z + 3 = -45 \]

To isolate the variable term, we need to get rid of the constant term on the same side as the variable. Here, we have a +3. By subtracting 3 from both sides:

\[ 6z + 3 - 3 = -45 - 3 \]

We effectively cancel out the +3, leaving us with:

\[ 6z = -48 \]

This process ensures that we have the variable term by itself on one side, simplifying the problem.
Solve for the Variable
Once the variable term is isolated, the next step is to solve for the variable. This means determining the value of the variable that will satisfy the equation.

In our example, we have:

\[ 6z = -48 \]

To solve for \( z \), we need to get rid of the coefficient (in this case, 6) that multiplies the variable. We do this by dividing both sides of the equation by 6:

\[ \frac{6z}{6} = \frac{-48}{6} \]

This simplifies to:

\[ z = -8 \]

Now, we have found the value of \( z \) that satisfies the equation.
Check the Solution
The final step is to verify that our solution is correct. We do this by substituting the value of the variable back into the original equation and ensuring that both sides are equal.

Starting with our original equation:

\[ 6z + 3 = -45 \]

We substitute \( z = -8 \):

\[ 6(-8) + 3 = -45 \]

Simplifying the left-hand side:

\[ -48 + 3 = -45 \]

We see that:

\[ -45 = -45 \]

This shows that the left-hand side equals the right-hand side, confirming that our solution \( z = -8 \) is indeed correct. Always check your solutions to ensure accuracy and to confirm that no mistakes were made during calculations.

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Most popular questions from this chapter

Insert the symbol \(<,>, \leq\) or \(\geq\) to make each pair of inequalities equivalent. $$ -7 t \geq 56 ; t \square-8 $$

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Solve. What percent of 300 is \(57 ?\)

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