Since the object in question is a cube, each of its sides must be the same length. Therefore, to get a volume of \(\displaystyle 216\) \(\displaystyle cm^{3}\), each side must be equal to the cube root of \(\displaystyle 216\), which is \(\displaystyle 6\) cm.

We can then set each expression equal to \(\displaystyle 6\).

The first expression \(\displaystyle (3x^2^\) \(\displaystyle -6)\) can be solved by either \(\displaystyle -2\) or \(\displaystyle 2\), but the other two expressions make it evident that the solution is \(\displaystyle x = -2\).

The most simple method for solving systems of equations is to transform one of the equations so it allows for the canceling out of a variable. In this case, we can multiply \(\displaystyle 3x + y = 8\) by \(\displaystyle (-4)\) to get \(\displaystyle -12x - 4y = -32\).

 Then, we can add \(\displaystyle 2x + 4y = 12\) to this equation to yield \(\displaystyle -10x = -20\), so \(\displaystyle x = 2\).

We can plug that value into either of the original equations; for example, \(\displaystyle 3(2) )+ y = 8\).

So, \(\displaystyle y = 2\) as well.

By solving one equation for \(\displaystyle y\), and replacing \(\displaystyle y\) in the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

Multiply the bottom equation by 5, then add to the top equation:

\(\displaystyle 6x-y = 24\)

\(\displaystyle 5 \left (6x-y \right ) =5\cdot 24\)

\(\displaystyle 30x-5y =120\)

\(\displaystyle \underline{\textrm{\; } 3x + 5y = \; \; 1}\)

\(\displaystyle 33x \;\;\;\;\;\; \; =121\)

\(\displaystyle x = \frac{121}{33} = \frac{11}{3} = 3 \frac{2}{3}\)

Multiply the top equation by \(\displaystyle -2\):

\(\displaystyle 3x + 5y = 23\)

\(\displaystyle -2 \cdot \left (3x + 5y \right )= -2 \cdot 23\)

\(\displaystyle -6x -10y \right )= -46\)

Now add:

   \(\displaystyle 6x-\; \; y = -9\)

\(\displaystyle \underline{-6x -10y = -46}\)

\(\displaystyle -11y = - 55\)

\(\displaystyle -11y \div (-11) = -55\div (-11)\)

\(\displaystyle y = 5\)

Multiply the top equation by \(\displaystyle -2\):

\(\displaystyle 3x + 5y = 1\)

\(\displaystyle -2 \cdot \left (3x + 5y \right )= -2 \cdot 1\)

\(\displaystyle -6x -10y \right )= -2\)

Now add:

    \(\displaystyle \; \; 6x-\; \; \; y = 24\)

\(\displaystyle \underline{-6x -10y= -2} \right )\)

          \(\displaystyle -11y = 22\)

\(\displaystyle -11y \div (-11) = 22\div (-11)\)

\(\displaystyle y = -2\)

\(\displaystyle a + 3b = 5\)

\(\displaystyle a-2b = 0\)

To solve this system of equations, use substitution. First, convert the second equation to isolate \(\displaystyle \small a\).

\(\displaystyle a-2b = 0\rightarrow a=2b\)

Then, substitute \(\displaystyle \small 2b\) into the first equation for \(\displaystyle \small a\).

\(\displaystyle a + 3b = 5\)

\(\displaystyle (2b) + 3b = 5\)

Combine terms and solve for \(\displaystyle \small b\).

\(\displaystyle 5b=5\)

\(\displaystyle b=1\)

Now that we know the value of \(\displaystyle \small b\), we can solve for \(\displaystyle \small a\) using our previous substitution equation.

\(\displaystyle a=2b=2(1)=2\)

When we add the two equations, the \(\displaystyle x\) and \(\displaystyle y\) variables cancel leaving us with:

\(\displaystyle 1=8\)   which means there is no solution for this system.

First, combine like terms to get \(\displaystyle 4x+3=7x+12\). Then, subtract 12 and \(\displaystyle 4x\) from both sides to separate the integers from the \(\displaystyle x\)'s to get \(\displaystyle -9=3x\). Finally, divide both sides by 3 to get \(\displaystyle x=-3\).

We are given the following system of equations:

\(\displaystyle y=-4x+8\)

\(\displaystyle y=3x-5\)

We are to find \(\displaystyle x\) and \(\displaystyle y\). We can solve this through the substitution method.  First, substitute the second equation into the first equation to get

\(\displaystyle 3x-5=-4x+8\)

Solve for \(\displaystyle x\) by adding 4x to both sides

\(\displaystyle 3x+4x-5=-4x+4x+8\)

\(\displaystyle 7x-5=8\)

Add 5 to both sides

\(\displaystyle 7x-5+5=8+5\)

\(\displaystyle 7x=13\)

Divide by 7

\(\displaystyle \frac{7x}{7}=\frac{13}{7}\)

\(\displaystyle x=\frac{13}{7}\)

So \(\displaystyle x=\frac{13}{7}\). Use this value to find \(\displaystyle y\) using one of the equations from our given system of equations.  I think I'll use the first equation (can also use the second equation).

\(\displaystyle y=-4x+8\)

\(\displaystyle y=-4\left (\frac{13}{7} \right )+8\)

\(\displaystyle y=-\frac{52}{7}+8\)

\(\displaystyle y=-\frac{52}{7}+\frac{56}{7}\)

\(\displaystyle y=\frac{4}{7}\)

So the two linear functions intersect at

\(\displaystyle (x,y)=\left(\frac{13}{7},\frac{4}{7} \right )\)