Question

current objective
solve a factored quadratic equation using the zero product property

question
solve: \\((10y - 6)(7y - 6) = 0\\). multiple answers, if any, should be separated by commas.

provide your answer below:
\\(y =\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Zero Product Property",
"Solving Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Zero Product Property",
"Solving Linear Equations"
]
}
</pre_analysis>

<reasoning>

Apply the Zero Product Property

Using the Zero Product Property knowledge point
\[
(10y - 6)(7y - 6) = 0 \implies 10y - 6 = 0 \quad \text{or} \quad 7y - 6 = 0
\]

Solve the first linear equation

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} 10y - 6 &= 0 \\ 10y &= 6 \\ y &= \frac{6}{10} = \frac{3}{5} \end{aligned}$$

\]

Solve the second linear equation

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} 7y - 6 &= 0 \\ 7y &= 6 \\ y &= \frac{6}{7} \end{aligned}$$

\]

Combine the solutions

The solutions to the equation are:
\[
y = \frac{3}{5}, \frac{6}{7}
\]
</reasoning>

<answer>
Solve: \((10y - 6)(7y - 6) = 0\). Multiple answers, if any, should be separated by commas. Enter your answer below:

\(y =\) <blank>\(\frac{3}{5}, \frac{6}{7}\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Zero Product Property"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Zero Product Property",
"Solving Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Zero Product Property",
"Solving Linear Equations"
]
}
</pre_analysis>

<reasoning>

Apply the Zero Product Property

Using the Zero Product Property knowledge point
\[
(10y - 6)(7y - 6) = 0 \implies 10y - 6 = 0 \quad \text{or} \quad 7y - 6 = 0
\]

Solve the first linear equation

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} 10y - 6 &= 0 \\ 10y &= 6 \\ y &= \frac{6}{10} = \frac{3}{5} \end{aligned}$$

\]

Solve the second linear equation

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} 7y - 6 &= 0 \\ 7y &= 6 \\ y &= \frac{6}{7} \end{aligned}$$

\]

Combine the solutions

The solutions to the equation are:
\[
y = \frac{3}{5}, \frac{6}{7}
\]
</reasoning>

<answer>
Solve: \((10y - 6)(7y - 6) = 0\). Multiple answers, if any, should be separated by commas. Enter your answer below:

\(y =\) <blank>\(\frac{3}{5}, \frac{6}{7}\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Zero Product Property"
]
}
</post_analysis>