Answer :
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.
### 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.
