Solve the system by substitution.y, equals, 6, xy=6xy, equals, 3, x, plus, 21y=3x+21

To solve the system of equations by substitution, we'll follow the steps below:

1. Break Down the Problem

We have the following system of equations:

1. $y = 6$
2. $6xy = 3x + 21$

2. Relevant Concepts

We'll substitute $y$ from the first equation into the second equation.

3. Analysis and Detail

Substituting $y = 6$ into the second equation $6xy = 3x + 21$:

$6x(6) = 3x + 21$ $36x = 3x + 21$

Now, isolate $x$ by moving $3x$ to the left side:

$36x - 3x = 21$ $33x = 21$

Dividing both sides by 33:

$x = \frac{21}{33} = \frac{7}{11}$

4. Verify and Summarize

We have $x = \frac{7}{11}$ and already know $y = 6$. Check by substituting back in the equations to ensure no errors occurred.

Substitute back into $6xy$:

$6 \left(\frac{7}{11}\right)(6) = 3\left(\frac{7}{11}\right) + 21$ $\frac{252}{11} = \frac{21}{11} + 21$ Converting $21$ to a fraction gives us $\frac{231}{11}$: $\frac{252}{11} = \frac{21 + 231}{11} = \frac{252}{11}$ Both sides equal, confirming the solution.

Final Answer

The solution to the system is: $\left( x, y \right) = \left( \frac{7}{11}, 6 \right)$