Three numbers are in arithmetic progression. Their sum is 3 and the sum of their squares is 11. What are the three numbers? I got one of the answers which is 5,7,9. What are the other three numbers?

The three numbers are -1, 1, and 3

Explanation:

The three terms sum to

first + second + third = (x)+(x+d)+(x+2d) = 3x+3d

We're told this sum is 3, so,

3x+3d = 3

3(x+d) = 3

x+d = 3/3

x+d = 1

x = 1-d

The three terms x, x+d, x+2d have the squares of x^2, (x+d)^2, and (x+2d)^2 in that order.

Add up those squares and set the sum equal to 11

x^2 + (x+d)^2 + (x+2d)^2 = 11

Then plug in x = 1-d and solve for d.

x^2 + (x+d)^2 + (x+2d)^2 = 11

(1-d)^2 + (1-d+d)^2 + (1-d+2d)^2 = 11

(1-d)^2 + (1)^2 + (1+d)^2 = 11

(1-2d+d^2) + 1 + (1+2d+d^2) = 11

2d^2+3 = 11

2d^2 = 11-3

2d^2 = 8

d^2 = 8/2

d^2 = 4

d = sqrt(4) or d = -sqrt(4)

d = 2 or d = -2

If d = 2, then,

One possible answer is to have the sequence -1, 1, 3

Summing the original terms gets us -1+1+3 = 3

Summing their squares gets us (-1)^2 + (1)^2 + (3)^2 = 1+1+9 = 11

We've confirmed this answer.

If d = -2, then,

We get the same terms, just in the reverse order.

The sequence overall is still the same when we write the numbers from smallest to largest.

So there's only one solution and it's {-1, 1, 3}

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3