Solving a System of 3 Equations with 7 Variables

[anemone]

Let $x_1,\, x_2,\,\cdots,\,x_7$ be real numbers such that

$x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\\4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\\9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123.$

Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.

 

[jvicens]

Thank you for sharing these equations. It seems that these equations are a system of linear equations with 7 unknowns. In order to find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$, we need to solve for the values of $x_1,\, x_2,\,\cdots,\,x_7$.

I will use the method of elimination to solve this system of equations. First, I will multiply the first equation by 4, the second equation by -1, and the third equation by -9. This will give us the following system of equations:

$4x_1+16x_2+36x_3+64x_4+100x_5+144x_6+196x_7=4\\-4x_1-9x_2-16x_3-25x_4-36x_5-49x_6-64x_7=-12\\-81x_1-144x_2-225x_3-324x_4-441x_5-576x_6-729x_7=-1107.$

Next, I will add the first and second equations to eliminate $x_1$:

$7x_2+20x_3+39x_4+64x_5+95x_6+132x_7=-8.$

Then, I will add the first and third equations to eliminate $x_1$ again:

$60x_2+189x_3+360x_4+625x_5+1080x_6+1701x_7=-1103.$

Now, we have a system of 2 equations with 2 unknowns: $7x_2+20x_3+39x_4+64x_5+95x_6+132x_7=-8$ and $60x_2+189x_3+360x_4+625x_5+1080x_6+1701x_7=-1103.$

Solving this system, we get the values of $x_2,\,x_3,\,x_4,\,x_5,\,x_6,\,x_