At which point do the graphs of the following equations intersect each other?
2x−y+4=0
x+y=-5
(1,1)
(-3,-2)
(2,-1)
(4,-5)
To find where the graphs intersect, we need to find a point that satisfies both equations. One way to do this is to solve for one variable in one equation and substitute it into the other equation:
From the second equation, we can solve for y:
y = -x - 5
Substituting this into the first equation:
2x - (-x-5) + 4 = 0
2x + x + 1 = 0
3x = -1
x = -1/3
Substituting x = -1/3 into the equation y = -x - 5:
y = -(-1/3) - 5 = -14/3
So the graphs intersect at the point (-1/3, -14/3). None of the given points match this, so it is not one of the answer choices.
To find the point of intersection between two equations, we can solve the system of equations simultaneously. Let's solve the given system of equations:
1.
2x − y + 4 = 0
2.
x + y = -5
To solve the system, we will use the method of substitution:
Step 1: Solve equation 2 for y:
x + y = -5
y = -x - 5
Step 2: Substitute the value of y from equation 2 into equation 1:
2x - (-x - 5) + 4 = 0
2x + x + 5 + 4 = 0
3x + 9 = 0
3x = -9
x = -3
Step 3: Substitute the value of x into the equation for y:
y = -x - 5
y = -(-3) - 5
y = 3 - 5
y = -2
Therefore, the point of intersection is (-3, -2).
Now let's check if the given points on the list satisfy both equations:
1. (1,1):
For equation 1:
2(1) - 1 + 4 = 2 - 1 + 4 = 5, which is not equal to 0.
For equation 2:
1 + 1 = 2, which is not equal to -5.
2. (-3,-2):
For equation 1:
2(-3) - (-2) + 4 = -6 + 2 + 4 = 0.
For equation 2:
-3 + (-2) = -5, which is equal to -5.
Since the values for equation 1 and equation 2 are satisfied for (-3,-2), the point (-3,-2) is the point of intersection for the given equations.