Let's analyze each given point to determine if it is a solution to the equation [tex]\( |x-2| - y = 1 \)[/tex].
### Given Points:
1. [tex]\((9, 6)\)[/tex]
2. [tex]\(\left(\frac{1}{7}, \frac{6}{7}\right)\)[/tex]
3. [tex]\((-5, 9)\)[/tex]
### Equation:
[tex]\[ |x-2| - y = 1 \][/tex]
#### Step-by-Step Analysis:
### (a) Point [tex]\((9, 6)\)[/tex]:
1. For [tex]\( x = 9 \)[/tex], compute [tex]\( |x-2| \)[/tex]:
[tex]\[
|9-2| = |7| = 7
\][/tex]
2. Subtract [tex]\( y \)[/tex] from the result:
[tex]\[
7 - 6 = 1
\][/tex]
3. We found that [tex]\( |9-2| - 6 = 1 \)[/tex]. Hence, [tex]\((9, 6)\)[/tex] satisfies the equation. Therefore, [tex]\((9, 6)\)[/tex] is a solution.
### (b) Point [tex]\(\left(\frac{1}{7}, \frac{6}{7}\right)\)[/tex]:
1. For [tex]\( x = \frac{1}{7} \)[/tex], compute [tex]\( |x-2| \)[/tex]:
[tex]\[
\left| \frac{1}{7} - 2 \right| = \left| \frac{1}{7} - \frac{14}{7} \right| = \left| -\frac{13}{7} \right| = \frac{13}{7}
\][/tex]
2. Subtract [tex]\( y \)[/tex] from the result:
[tex]\[
\frac{13}{7} - \frac{6}{7} = \frac{13-6}{7} = \frac{7}{7} = 1
\][/tex]
3. We found that [tex]\( \left| \frac{1}{7} - 2 \right| - \frac{6}{7} = 1 \)[/tex]. Hence, [tex]\(\left(\frac{1}{7}, \frac{6}{7}\right)\)[/tex] satisfies the equation. Therefore, [tex]\((\frac{1}{7}, \frac{6}{7})\)[/tex] is a solution.
### (c) Point [tex]\((-5, 9)\)[/tex]:
1. For [tex]\( x = -5 \)[/tex], compute [tex]\( |x-2| \)[/tex]:
[tex]\[
|-5 - 2| = |-7| = 7
\][/tex]
2. Subtract [tex]\( y \)[/tex] from the result:
[tex]\[
7 - 9 = -2
\][/tex]
3. We found that [tex]\( |-5-2| - 9 = -2 \)[/tex], which does not equal 1. Hence, [tex]\((-5, 9)\)[/tex] does not satisfy the equation. Therefore, [tex]\((-5, 9)\)[/tex] is not a solution.
### Summary:
- [tex]\((9, 6)\)[/tex] is a solution to the equation [tex]\( |x-2| - y = 1 \)[/tex].
- [tex]\(\left(\frac{1}{7}, \frac{6}{7}\right)\)[/tex] is a solution to the equation [tex]\( |x-2| - y = 1 \)[/tex].
- [tex]\((-5, 9)\)[/tex] is not a solution to the equation [tex]\( |x-2| - y = 1 \)[/tex].
So the results are:
(a) [tex]\((9,6)\)[/tex] is a solution.
(b) [tex]\(\left(\frac{1}{7}, \frac{6}{7}\right)\)[/tex] is a solution.
(c) [tex]\((-5,9)\)[/tex] is not a solution.
