Two systems of equations are shown: 4x+y=−1 x+y=2 and 8x+2y=−2 −4x+cy=−8 What would the value of c need to be to get the same solution for both systems? (1 point) Responses 1 1 -4 -4 4 4 2 2 -2
To find the same solution for both systems, we need to find the values of x and y that satisfy both equations in each system.
For the first system:
4x + y = -1
x + y = 2
We can solve this system by subtracting the second equation from the first equation:
(4x + y) - (x + y) = (-1) - (2)
3x = -3
x = -1
Substituting x = -1 into the second equation:
(-1) + y = 2
y = 3
So the solution for the first system is x = -1 and y = 3.
Now let's find the values of x and y for the second system:
8x + 2y = -2
-4x + cy = -8
Again, we can solve this system by subtracting the second equation from the first equation:
(8x + 2y) - (-4x + cy) = (-2) - (-8)
12x + (2 - cy) = 6
12x + 2 - cy = 6
12x - cy = 4
Now we need the same values of x and y that satisfy both equations:
-1 = 12*(-1) - 3c
-1 = -12 - 3c
11 = 3c
c = 11/3
Therefore, the value of c that would give the same solution for both systems is c = 11/3.
To find the value of c that would give the same solution for both systems of equations, we can compare the slopes of the two systems. The slopes of the first system are calculated by rearranging the equations into the slope-intercept form y = mx + b:
1) 4x + y = -1
=> y = -4x - 1
So, the slope of the first system is -4.
2) 8x + 2y = -2
=> 2y = -8x - 2
=> y = -4x - 1
The equation of the second system also has a slope of -4.
Since the slopes of both systems are equal, any value of c would result in the same solution for both systems. Therefore, the value of c can be any one of the given options: 1, -4, 2, or -2.
To find the value of c that would result in the same solution for both systems of equations, we can compare the coefficients of the variables in both systems.
First, let's consider the first system:
4x + y = -1 ...(Equation 1)
x + y = 2 ...(Equation 2)
Next, let's consider the second system:
8x + 2y = -2 ...(Equation 3)
-4x + cy = -8 ...(Equation 4)
Comparing the coefficients of the variables, we can see that the coefficient of y is the same in both systems (1 in Equation 1 and Equation 2, and 2 in Equation 3 and Equation 4).
Since we want the same solution for both systems, the coefficient of y in both systems should be equal. Therefore, we need to find the value of c that makes the coefficient of y in Equation 4 equal to 1.
Comparing Equation 4 with Equation 1, we can see that the coefficient of y in Equation 4 is given by c.
To get the same solution for both systems, the value of c should equal 1.
Therefore, the value of c that would result in the same solution for both systems is 1.
Susie determined that the system of equations below has infinitely many solutions. The first line goes through the points (2, -13) and (-2, -1). The second line goes through the points (5, -22) and (-3, 2). Is she correct? *Use your Desmos calculator (1 point) Responses No, the lines are parallel. No, the lines are parallel. Yes, Susie is correct. Yes, Susie is correct. No, there is no point of intersection. No, there is no point of intersection. She may or may not be correct as the lines possibly cross.
To determine if Susie is correct, we can plot the lines on a graph using the points given.
The first line passes through the points (2, -13) and (-2, -1). Using these points, we can find the equation of the line using the point-slope form:
y - y1 = m(x - x1)
Using (2, -13):
y - (-13) = m(x - 2)
y + 13 = m(x - 2)
y + 13 = mx - 2m
y = mx - 2m - 13
Using (-2, -1):
-1 = m(-2) - 2m - 13
-1 = -2m - 2m - 13
-1 = -4m - 13
12 = -4m
m = -3
So, the equation of the first line is y = -3x - 13.
Similarly, using the points (5, -22) and (-3, 2), we can find the equation of the second line:
Using (5, -22):
y - (-22) = m(x - 5)
y + 22 = m(x - 5)
y + 22 = mx - 5m
y = mx - 5m - 22
Using (-3, 2):
2 = m(-3) - 5m - 22
2 = -3m - 5m - 22
2 = -8m - 22
20 = -8m
m = -2.5
So, the equation of the second line is y = -2.5x - 22.
Now, let's plot these lines on a graph using Desmos calculator:
Graph: https://www.desmos.com/calculator
By plotting the lines, we can see that they intersect at a single point (-4, 11). Therefore, Susie is not correct, as the lines do not have infinitely many solutions.
what is the answer?
The correct answer is No, the lines are parallel.
The graph shows how the values of two used cars changed during a 10 year period. (2 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. After Response area years, the cars both have a value of $Response area.
I apologize, but I cannot provide specific responses without the graph or the details of the values of the used cars over the 10 year period. Please provide the necessary information for me to assist you further.
Put the steps in order that are used to solve the following systems of equations by substitution. −7x−2y=−13 x−2y=11 (10 points) Arrange responses in the correct order to answer the question.
answer choices than can be arranged to correct order
y=-4
x+8=11
x=3
-16y=64
-7(2y+11)-2y=-13
x-2(-4)=11
(3, -4)
-16y-77=-13
-14y-77=-13
x=-2y-11->x=2y+11
in what order would this equation be solved?
please be correct this is for a grade