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
