Solve the system graphically.
-$$\begin{array} { l }
x ^ { 2 } - y ^ { 2 } = 7.82 \
x = 2 y - 5 \
\begin{aligned}
\text {

A) } \approx ( - 2.97 , - 1.01 ) \text { and } \approx ( 6.31 , - 5.65 ) & \text {

B) } \approx ( - 6.31,5.65 ) \text { and } \approx ( 2.97,1.01 ) \
\text {

C) } \approx ( - 6.31 , - 5.65 ) \text { and } \approx ( 2.97 , - 1.01 ) & \text {

D) } \approx ( - 2.97,1.01 ) \text { and } \approx ( 6.31,5.65 )
\end{aligned}
\end{array}$$

Question

Solve the system graphically.
-$$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 7.82 \
x = 2 y - 5 \
\begin{aligned}
\text {

A) } \approx ( - 2.97 , - 1.01 ) \text { and } \approx ( 6.31 , - 5.65 ) & \text {

B) } \approx ( - 6.31,5.65 ) \text { and } \approx ( 2.97,1.01 ) \
\text {

C) } \approx ( - 6.31 , - 5.65 ) \text { and } \approx ( 2.97 , - 1.01 ) & \text {

D) } \approx ( - 2.97,1.01 ) \text { and } \approx ( 6.31,5.65 )
\end{aligned}
\end{array}$$

Quizplus · Accepted Answer

To solve the system graphically, we need to plot both equations on the same set of axes and find their points of intersection.
The first equation is a hyperbola:
$$ x^2 - y^2 = 7.82 $$
The second equation is a line:
$$ x = 2y - 5 $$
Let's rearrange the second equation to solve for y:
$$ y = \frac{x + 5}{2} $$
Now, we can plot both the hyperbola and the line on a graph. The points of intersection will be the solutions to the system.
Since we are looking for an approximate graphical solution, we can use the given options to check which one fits the system of equations.
Let's substitute the x and y values from each option into both equations to see which one satisfies both.
Option A:
For the point (-2.97, -1.01):
$$ x^2 - y^2 = (-2.97)^2 - (-1.01)^2 = 8.82 - 1.02 \approx 7.80 $$
$$ x = 2y - 5 = 2(-1.01) - 5 = -2.02 - 5 \approx -7.02 $$
This does not match the given x value of -2.97.
For the point (6.31, -5.65):
$$ x^2 - y^2 = (6.31)^2 - (-5.65)^2 = 39.80 - 31.92 \approx 7.88 $$
$$ x = 2y - 5 = 2(-5.65) - 5 = -11.30 - 5 \approx -16.30 $$
This does not match the given x value of 6.31.
Option B:
For the point (-6.31, 5.65):
$$ x^2 - y^2 = (-6.31)^2 - (5.65)^2 = 39.80 - 31.92 \approx 7.88 $$
$$ x = 2y - 5 = 2(5.65) - 5 = 11.30 - 5 \approx 6.30 $$
This does not match the given x value of -6.31.
For the point (2.97, 1.01):
$$ x^2 - y^2 = (2.97)^2 - (1.01)^2 = 8.82 - 1.02 \approx 7.80 $$
$$ x = 2y - 5 = 2(1.01) - 5 = 2.02 - 5 \approx -2.98 $$
This does not match the given x value of 2.97.
Option C:
For the point (-6.31, -5.65):
$$ x^2 - y^2 = (-6.31)^2 - (-5.65)^2 = 39.80 - 31.92 \approx 7.88 $$
$$ x = 2y - 5 = 2(-5.65) - 5 = -11.30 - 5 \approx -16.30 $$
This does not match the given x value of -6.31.
For the point (2.97, -1.01):
$$ x^2 - y^2 = (2.97)^2 - (-1.01)^2 = 8.82 - 1.02 \approx 7.80 $$
$$ x = 2y - 5 = 2(-1.01) - 5 = -2.02 - 5 \approx -7.02 $$
This does not match the given x value of 2.97.
Option D:
For the point (-2.97, 1.01):
$$ x^2 - y^2 = (-2.97)^2 - (1.01)^2 = 8.82 - 1.02 \approx 7.80 $$
$$ x = 2y - 5 = 2(1.01) - 5 = 2.02 - 5 \approx -2.98 $$
This does not match the given x value of -2.97.
For the point (6.31, 5.65):
$$ x^2 - y^2 = (6.31)^2 - (5.65)^2 = 39.80 - 31.92 \approx 7.88 $$
$$ x = 2y - 5 = 2(5.65) - 5 = 11.30 - 5 \approx 6.30 $$
This does not match the given x value of 6.31.
None of the options provided exactly match the solutions when plugged into the equations. However, since we are looking for approximate solutions, we should choose the option that comes closest to satisfying both equations. Based on the calculations, Option A seems to be the closest, with slight discrepancies likely due to rounding errors in the approximation.
Therefore, the answer is:
A) Approximately (-2.97, -1.01) and (6.31, -5.65)

Solve the System Graphically