Sure, let's analyze each pair of lines step-by-step to determine whether they are parallel, perpendicular, or neither.
### Question 25: [tex]\(6x + 10y = 20\)[/tex] and [tex]\(5x - 3y = 21\)[/tex]
Step 1: Find the slope of the first equation, [tex]\(6x + 10y = 20\)[/tex]
Rewriting the equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 6x + 10y = 20 \][/tex]
[tex]\[ 10y = -6x + 20 \][/tex]
[tex]\[ y = \frac{-6}{10}x + 2 \][/tex]
The slope ([tex]\(m_1\)[/tex]) is [tex]\(\frac{-6}{10}\)[/tex], which simplifies to [tex]\(-0.6\)[/tex].
Step 2: Find the slope of the second equation, [tex]\(5x - 3y = 21\)[/tex]
Rewriting the equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 5x - 3y = 21 \][/tex]
[tex]\[ -3y = -5x + 21 \][/tex]
[tex]\[ y = \frac{5}{3}x - 7 \][/tex]
The slope ([tex]\(m_2\)[/tex]) is [tex]\(\frac{5}{3}\)[/tex], which is approximately 1.67.
Step 3: Compare the slopes
The slopes of the lines are [tex]\(-0.6\)[/tex] and [tex]\(\frac{5}{3}\)[/tex] (approximately 1.67). Since these slopes are neither equal nor negative reciprocals of each other, the lines are neither parallel nor perpendicular.
### Conclusion for Question 25
The lines [tex]\(6x + 10y = 20\)[/tex] and [tex]\(5x - 3y = 21\)[/tex] are neither parallel nor perpendicular.
### Question 26: [tex]\(x - y = 4\)[/tex] and [tex]\(x + y = 9\)[/tex]
Step 1: Find the slope of the first equation, [tex]\(x - y = 4\)[/tex]
Rewriting the equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ x - y = 4 \][/tex]
[tex]\[ -y = -x + 4 \][/tex]
[tex]\[ y = x - 4 \][/tex]
The slope ([tex]\(m_1\)[/tex]) is [tex]\(1\)[/tex].
Step 2: Find the slope of the second equation, [tex]\(x + y = 9\)[/tex]
Rewriting the equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ x + y = 9 \][/tex]
[tex]\[ y = -x + 9 \][/tex]
The slope ([tex]\(m_2\)[/tex]) is [tex]\(-1\)[/tex].
Step 3: Compare the slopes
The slopes of the lines are [tex]\(1\)[/tex] and [tex]\(-1\)[/tex]. Since the product of these slopes is [tex]\(-1\)[/tex] (1 * -1 = -1), the lines are perpendicular.
### Conclusion for Question 26
The lines [tex]\(x - y = 4\)[/tex] and [tex]\(x + y = 9\)[/tex] are perpendicular.
### Summary
- For lines [tex]\(6x + 10y = 20\)[/tex] and [tex]\(5x - 3y = 21\)[/tex], they are neither parallel nor perpendicular.
- For lines [tex]\(x - y = 4\)[/tex] and [tex]\(x + y = 9\)[/tex], they are perpendicular.
