Sure, let's solve each of these equations step-by-step and find the respective lines that intersect with the graph of [tex]\( y = x^2 - 8x - 7 \)[/tex].
### a) [tex]\( x = 8 + \frac{7}{x} \)[/tex]
To find the corresponding line equation:
1. Multiply both sides by [tex]\( x \)[/tex] to eliminate the fraction:
[tex]\[ x^2 = 8x + 7 \][/tex]
2. Rearrange to form a standard quadratic equation:
[tex]\[ x^2 - 8x - 7 = 0 \][/tex]
3. To solve [tex]\( x = 8 + \frac{7}{x} \)[/tex], we need the points of intersection with [tex]\( y = x \)[/tex]. Hence, maintain:
[tex]\[ y = 8 + \frac{7}{x} \][/tex]
Thus, the equation of the line to draw is:
[tex]\[ y = 8 + \frac{7}{x} \][/tex]
### b) [tex]\( 2x^2 = 16x + 9 \)[/tex]
To find the corresponding line equation:
1. Divide by 2 to simplify:
[tex]\[ x^2 = 8x + \frac{9}{2} \][/tex]
2. Rearrange to:
[tex]\[ x^2 - 8x - \frac{9}{2} = 0 \][/tex]
3. To solve [tex]\( 2x^2 = 16x + 9 \)[/tex], we need the points of intersection with [tex]\( y = 2x^2 \)[/tex]. Hence, maintain:
[tex]\[ y = 16x + 9 \][/tex]
So, the equation of the line to draw is:
[tex]\[ y = 8x + \frac{9}{2} \][/tex]
### c) [tex]\( x^2 = 7 \)[/tex]
To find the corresponding line equation:
1. This is already solved:
[tex]\[ x^2 - 7 = 0 \][/tex]
2. The parabolic form intersecting [tex]\( y = x^2 \)[/tex] remains straightforward:
Thus, it's:
[tex]\[ y = 7 \][/tex]
### d) [tex]\( x = \frac{4}{x-8} \)[/tex]
To find the corresponding line equation:
1. Multiply both sides by [tex]\( x-8 \)[/tex]:
[tex]\[ x(x-8) = 4 \][/tex]
2. Expand and rearrange:
[tex]\[ x^2 - 8x - 4 = 0 \][/tex]
3. To solve [tex]\( x = \frac{4}{x-8} \)[/tex], maintain:
[tex]\[ y = \frac{4}{x-8} \][/tex]
So, the equation of the line to draw is:
[tex]\[ y = \frac{4}{x-8} \][/tex]
### e) [tex]\( 2x - 5 = \frac{14}{x} \)[/tex]
To find the corresponding line equation:
1. Multiply both sides by [tex]\( x \)[/tex]:
[tex]\[ 2x^2 - 5x = 14 \][/tex]
2. Rearrange:
[tex]\[ 2x^2 - 5x - 14 = 0 \][/tex]
3. To solve [tex]\( 2x - 5 = \frac{14}{x} \)[/tex], maintain:
[tex]\[ y = 2x - 5 + \frac{14}{x} \][/tex]
So, the equation of the line to draw is:
[tex]\[ y = 2x - 5 + \frac{14}{x} \][/tex]
These lines, when graphed alongside [tex]\( y = x^2 - 8x - 7 \)[/tex], will visually represent the solutions to the given equations by their points of intersection.
