Answered

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Question 5 of 5

Select the correct answer.

Given: Prove [tex]\( m_1 = m_4 \)[/tex]

Step 1: Represent each line with a linear equation:
[tex]\[
\begin{array}{l}
y_1 = m_1 x + b_1 \\
y_4 = m_4 x + b_4
\end{array}
\][/tex]

Step 2: Lines that are parallel do not intersect, so they have no points in common. If you try to solve for [tex]\( x \)[/tex] in parallel lines, there will be no solution. Use the equations to solve for [tex]\( x \)[/tex]:
[tex]\[
\begin{array}{c}
m_1 x + b_1 = m_4 x + b_4 \\
m_1 x - m_4 x = b_4 - b_1 \\
x(m_1 - m_4) = b_1 - b_4
\end{array}
\][/tex]

Step 3: For [tex]\( x \)[/tex] to have no solution, [tex]\( b_1 - b_4 \)[/tex] must equal 0. Set [tex]\( b_1 - b_4 \)[/tex] equal to 0:
[tex]\[
\begin{array}{r}
b_1 - b_4 = 0 \\
b_1 = b_4
\end{array}
\][/tex]

Step 4: This gives us that [tex]\( m_1 = m_4 \)[/tex], proving the slopes of parallel lines must be equal.

In which step did Angelica make a mistake?

A. Step 2: Angelica forgot a negative sign when rearranging the equation [tex]\( b_1 - b_4 = 0 \)[/tex].

B. Step 2: Solving for parallel lines will always result in infinite solutions instead of no solution.

C. Step 2: Angelica made a mistake in rearranging the algebraic equation.

D. Step 3: The expression [tex]\( m_1 - m_4 \)[/tex] must equal 0 to have no solution instead of the y-intercepts.

Sagot :

GinnyAnswer
Let's carefully go through the steps provided and analyze them to determine where Angelica made a mistake.

### Step-by-Step Explanation:

Step 1: Represent each line with a linear equation:
[tex]\[ \begin{array}{l} y_1 = m_1 x + b_1 \\ y_4 = m_4 x + b_4 \end{array} \][/tex]

Step 2: Solve for [tex]\( x \)[/tex] assuming the lines are parallel (parallel lines have no common points and the slope-intercept form should reflect this). Set the equations equal to each other:
[tex]\[ \begin{array}{c} m_1 x + b_1 = m_4 x + b_4 \\ m_1 x - m_4 x = b_4 - b_1 \\ x (m_1 - m_4) = b_4 - b_1 \end{array} \][/tex]

Analysis of Step 2:
From the equation [tex]\( x (m_1 - m_4) = b_4 - b_1 \)[/tex], for parallel lines, [tex]\( m_1 \)[/tex] should equal [tex]\( m_4 \)[/tex]. This ensures the left-hand side of the equation becomes zero since [tex]\( m_1 - m_4 = 0 \)[/tex]. Hence the equation simplifies to [tex]\( 0 = b_4 - b_1 \)[/tex], i.e., [tex]\( b_4 - b_1 = 0 \)[/tex].

Next, we solve for [tex]\( b_4 \)[/tex] in terms of [tex]\( b_1 \)[/tex]:
[tex]\[ b_4 - b_1 = 0 \implies b_4 = b_1 \][/tex]

Step 3: Confirm that for [tex]\( x \)[/tex] to have no solution, [tex]\( b_4 - b_1 \)[/tex] must equal 0:
[tex]\[ \begin{array}{r} b_4 - b_1 = 0 \\ b_4 = b_1 \end{array} \][/tex]

Conclusion:
From supporting that [tex]\( b_4 = b_1 \)[/tex], we derive that for lines to be parallel, indeed [tex]\( m_1 = m_4 \)[/tex] (as slopes must be equal for parallel lines).

Step 4: Prove that [tex]\( b_1 = b_4 \)[/tex] confirms that the slopes [tex]\( m_1 \)[/tex] and [tex]\( m_4 \)[/tex] are equal.

### Identifying the Mistake:

Angelica made a mistake in Step 2 during the rearrangement of the algebraic equation. The correct form is [tex]\( b_4 - b_1 = 0 \)[/tex], where she should have set [tex]\( b_4 \)[/tex] equal to [tex]\( b_1 \)[/tex]. Hence the correct answer is:

Step 2: Angelica made a mistake in rearranging the algebraic equation.
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