To determine whether your friend's work makes sense, we need to analyze her solution and check if the lines [tex]\( y = 2x - 5 \)[/tex] and [tex]\( y = \frac{1}{2}x + 5 \)[/tex] are indeed perpendicular.
1. Calculate the slopes of the lines:
- The slope of the first line [tex]\( y = 2x - 5 \)[/tex]:
The equation is already in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Thus, the slope ([tex]\( m_1 \)[/tex]) is [tex]\( 2 \)[/tex].
- The slope of the second line [tex]\( y = \frac{1}{2}x + 5 \)[/tex]:
Similarly, this equation is in the form [tex]\( y = mx + b \)[/tex]. Therefore, the slope ([tex]\( m_2 \)[/tex]) is [tex]\( \frac{1}{2} \)[/tex].
2. Check the product of their slopes:
- For two lines to be perpendicular, the product of their slopes must be [tex]\( -1 \)[/tex].
- Calculate the product of the slopes:
[tex]\[
\text{Product of slopes} = m_1 \times m_2 = 2 \times \frac{1}{2} = 1
\][/tex]
3. Evaluate the result:
- The product of the slopes is [tex]\( 1 \)[/tex], not [tex]\( -1 \)[/tex]. Hence, the lines are not perpendicular.
4. Examine her algebraic steps briefly:
- While her algebraic manipulation has errors (particularly [tex]\( 6 = 1 + 3 \rightarrow 5 = 3 \)[/tex]), which are clearly incorrect, correcting these does not change the slopes of the lines.
Based on this analysis, her work does not make sense. The lines [tex]\( y = 2x - 5 \)[/tex] and [tex]\( y = \frac{1}{2}x + 5 \)[/tex] are not perpendicular because the product of their slopes is [tex]\( 1 \)[/tex], not [tex]\( -1 \)[/tex]. Therefore, the lines do not meet the perpendicularity condition.
