Solve the system x + 3y = 6 and x - (1/3)y = 0 for x and y.

Step-by-step solution

1. 1

   Pick the equation that isolates most cleanly. The second equation has $x$ with coefficient $1$ and no constant term, so solving it for $x$ costs nothing:

   $x-\frac13y=0\;\Longrightarrow\;x=\frac13y.$

   Choosing the other equation would work too, but this one produces the simplest expression to substitute.

2. 2

   Substitute into the first equation. Replacing $x$ by $\tfrac13y$ in $x+3y=6$:

   $\frac13y+3y=6.$

   Now there is a single unknown, which is the whole point of substitution.

3. 3

   Combine the $y$ terms over a common denominator. Writing $3y=\tfrac93y$:

   $\frac13y+\frac93y=\frac{10}{3}y=6.$

   The combined coefficient $\tfrac{10}{3}$ is not a whole number, which is exactly why the answers come out as fifths rather than integers.

4. 4

   Solve for $y$, then back-substitute for $x$. Multiplying by the reciprocal $\tfrac{3}{10}$:

   $y=6\cdot\frac{3}{10}=\frac{18}{10}=\frac95,$

   and then

   $x=\frac13\cdot\frac95=\frac{9}{15}=\frac35.$

   Both fractions should be reduced: $\tfrac{18}{10}\to\tfrac95$ and $\tfrac{9}{15}\to\tfrac35$.

5. 5

   Check in both original equations. First: $\tfrac35+3\cdot\tfrac95=\tfrac35+\tfrac{27}{5}=\tfrac{30}{5}=6$ ✓. Second: $\tfrac35-\tfrac13\cdot\tfrac95=\tfrac35-\tfrac35=0$ ✓. As decimals the solution is $(0.6,\,1.8)$, and the geometric reading is that the lines $x+3y=6$ and $y=3x$ cross there.