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Prove: [tex]m_c = m_d[/tex]

Select the correct answer from each drop-down menu.

Statements:
1. [tex]c \parallel d[/tex]
2. [tex]c = m_c x + b_c[/tex]
3. [tex]d = m_d x + b_d[/tex]
4.
[tex]\[
\begin{array}{l}
m_c x + b_c = m_d x + b_d \\
m_c x - m_d x = b_d - b_c \\
x(m_c - m_d) = b_d - b_c \\
x = \frac{b_d - b_c}{m_c - m_d} \\
m_c - m_d = 0 \\
m_c = m_d
\end{array}
\][/tex]

Reasons:
1. given
2. write equations for lines [tex]c[/tex] and [tex]d[/tex]
3. solve for [tex]x[/tex]
4. subtraction property of equality
5. distributive property
6. division property of equality
7. [tex]x[/tex] does not exist; therefore, the denominator must be 0
8. addition property of equality



Answer :

GinnyAnswer
Let's go through the detailed, step-by-step solution to prove \( m_c = m_d \):

1. Given:
[tex]\[ c \parallel d \][/tex]
Reason: given

2. Write equations for lines \( c \) and \( d \):
[tex]\[ c = m_c x + b_c \][/tex]
[tex]\[ d = m_d x + b_d \][/tex]
Reason: write equations for lines \( c \) and \( d \)

3. Set the equations equal to each other since they are parallel:
[tex]\[ m_c x + b_c = m_d x + b_d \][/tex]
Reason: addition property of equality

4. Subtract the terms involving \( b \) and \( m \) from both sides to isolate the \( x \) terms on one side:
[tex]\[ m_c x - m_d x = b_d - b_c \][/tex]
Reason: subtraction property of equality

5. Factor out \( x \) on the left-hand side:
[tex]\[ x (m_c - m_d) = b_d - b_c \][/tex]
Reason: distributive property

6. Solve for \( x \). However, parallel lines imply they never intersect, meaning \( x \) does not exist:
[tex]\[ x = \frac{b_d - b_c}{m_c - m_d} \][/tex]
Since \( x \) does not exist, the equation must be invalid, leading to an explanation:
Reason: division property of equality

7. Since \( x \) does not exist, the denominator must be 0:
[tex]\[ m_c - m_d = 0 \][/tex]
Reason: \( x \) does not exist; therefore, the denominator must be 0

8. Conclude from the last step:
[tex]\[ m_c = m_d \][/tex]
Reason: addition property of equality

So, to summarize:

Statements:
1. \(c \parallel d\)
2. \(c = m_c x + b_c \)
3. \(d = m_d x + b_d \)
4. \(m_c x + b_c = m_d x + b_d\)
5. \(m_c x - m_d x = b_d - b_c\)
6. \( x (m_c - m_d) = b_d - b_c \)
7. \( x = \frac{b_d - b_c}{m_c - m_d}\)
8. \(m_c - m_d = 0\)
9. \(m_c = m_d\)

Reasons:
1. given
2. write equations for lines \(c\) and \(d\)
3. addition property of equality
4. subtraction property of equality
5. distributive property
6. division property of equality
7. \( x \) does not exist; therefore, the denominator must be 0
8. addition property of equality

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