The graph of linear equation A passes through the points (−7,4)
and (3,−10)
, while the graph of linear equation B passes through the points (−7,4)
and (5,11)
. Which of these is a solution to the system of equations consisting of linear equation A and linear equation B?
A
(−7,4)
B
(3,−10)
C
(5,11)
D
(7,4)
The solution to a system of equations represents the point where the two equations intersect. Thus, the solution to the system of equations consisting of linear equation A and linear equation B is the point that is common to both equations.
We can determine the equation for line A using the two given points: (-7,4) and (3,-10). Using the formula for the equation of a line (y = mx + b), we can find the slope of line A:
m = (y2 - y1) / (x2 - x1)
m = (-10 - 4) / (3 - (-7))
m = -14 / 10
m = -7/5
Using the slope-intercept form of a line, we can find the y-intercept (b) of line A:
y = mx + b
4 = (-7/5)(-7) + b
4 = 49/5 + b
4 - 49/5 = b
20/5 - 49/5 = b
-29/5 = b
So, the equation for line A is y = (-7/5)x - 29/5.
Using the same process, we can find the equation for line B using the points (-7,4) and (5,11):
m = (11 - 4) / (5 - (-7))
m = 7/12
y = (7/12)x + b
4 = (7/12)(-7) + b
b = 4 + 49/12
b = 48/12 + 49/12
b = 97/12
Therefore, the equation for line B is y = (7/12)x + 97/12.
To find the point of intersection, we can set the two equations equal to each other and solve for x:
(-7/5)x - 29/5 = (7/12)x + 97/12
Multiplying both sides of the equation by 60 to eliminate the fractions:
-12(7)x - 12(29) = 5(7)x + 5(97)
-84x - 348 = 35x + 485
-119x = 833
x = -833/119
Substituting this value back into one of the equations, we can find the corresponding y-value:
y = (-7/5)(-833/119) - 29/5
y = 5831/595
Therefore, the point of intersection is approximately (-833/119, 5831/595).
None of the given points (−7,4), (3,−10), (5,11), or (7,4) match this point. Therefore, the solution to the system of equations is not one of the given points.
The correct answer is none of the above.