Solve the system x/y = 0.99 and x/(y + 23) = 0.98 for x and y.

Step-by-step solution

1. 1

   Clear both denominators. Multiplying each equation by its denominator turns two fractions into two linear relations:

   $x=0.99y,\qquad x=0.98\,(y+23).$

   Record the restrictions $y\ne0$ and $y\ne-23$; the answer will comfortably satisfy both. Working with fractions directly, or cross-multiplying between the two equations, is far more error-prone than this.

2. 2

   Equate the two expressions for $x$. Since both right-hand sides equal the same $x$,

   $0.99y=0.98\,(y+23).$

   This is the substitution method in its cleanest form: both equations were already solved for the same variable.

3. 3

   Expand and collect. Distributing the $0.98$:

   $0.99y=0.98y+22.54,$

   and subtracting $0.98y$ from both sides:

   $0.01y=22.54.$

   The tiny coefficient $0.01$ is the difference of the two ratios, which is why a modest shift of $23$ in the denominator produces such a large $y$: the answer scales like $\frac{23\times 0.98}{0.01}$.

4. 4

   Solve for $y$, then $x$. Dividing by $0.01$ (equivalently multiplying by $100$):

   $y=\frac{22.54}{0.01}=2254,$

   and then

   $x=0.99\times 2254=2231.46.$

5. 5

   Check both original ratios. First: $\frac{2231.46}{2254}=0.99$ ✓. Second: $y+23=2277$ and $\frac{2231.46}{2277}=0.98$ ✓. Verifying the second equation matters most, since it was the one used only indirectly — a slip in the $22.54$ would show up here and nowhere else.