Let's go through the provided steps for solving the equation [tex]\(8 - 3x = 20\)[/tex] and identify the missing step.
Step 1: [tex]\(8 - 3x - 8 = 20 - 8\)[/tex]
Justification: Subtraction property of equality.
Here, we subtract 8 from both sides of the equation to isolate the term involving [tex]\(x\)[/tex].
Step 2: [tex]\(-3x = 12\)[/tex]
Justification: Simplification.
Simplifying both sides of the equation, we get [tex]\(-3x = 12\)[/tex].
Now, we need to isolate [tex]\(x\)[/tex]. To do so, we need to divide both sides by the coefficient of [tex]\(x\)[/tex], which is [tex]\(-3\)[/tex].
The possible equations for the missing step (Step 3) are:
1. [tex]\(-3x \cdot \left(-\frac{1}{3}\right) = 12 \cdot \left(-\frac{1}{3}\right)\)[/tex]
2. [tex]\(-3x \cdot 3 = 12 \cdot 3\)[/tex]
3. [tex]\(-3x \cdot (-3) = 12 \cdot (-3)\)[/tex]
4. [tex]\(-3x \cdot \frac{1}{12} = 12 \cdot \frac{1}{12}\)[/tex]
Let's analyze:
- [tex]\(-3x \cdot \left(-\frac{1}{3}\right) = 12 \cdot \left(-\frac{1}{3}\right)\)[/tex]
Multiplying both sides by [tex]\(-\frac{1}{3}\)[/tex] effectively divides by [tex]\(-3\)[/tex], isolating [tex]\(x\)[/tex]:
[tex]\[
-3x \cdot \left(-\frac{1}{3}\right) = 12 \cdot \left(-\frac{1}{3}\right)
\][/tex]
- [tex]\(-3x \cdot 3 = 12 \cdot 3\)[/tex]
Multiplying by [tex]\(3\)[/tex] instead of dividing by [tex]\(-3\)[/tex] does not isolate [tex]\(x\)[/tex].
- [tex]\(-3x \cdot (-3) = 12 \cdot (-3)\)[/tex]
Multiplying by [tex]\(-3\)[/tex] instead of dividing by [tex]\(-3\)[/tex] does not isolate [tex]\(x\)[/tex].
- [tex]\(-3x \cdot \frac{1}{12} = 12 \cdot \frac{1}{12}\)[/tex]
Multiplying both sides by [tex]\(\frac{1}{12}\)[/tex] does not isolate [tex]\(x\)[/tex] as it’s not equivalent to dividing by [tex]\(-3\)[/tex].
Thus, the correct missing step is:
[tex]\[
-3x \cdot \left(-\frac{1}{3}\right) = 12 \cdot \left(-\frac{1}{3}\right)
\][/tex]
Hence, the correct answer to the question is:
[tex]\[
-3x \cdot \left(-\frac{1}{3}\right) = 12 \cdot \left(-\frac{1}{3}\right)
\][/tex]
