Solve the system: X * 5.20 + Y * 5.75 = 8947.2 and X + Y = 1595.

Step-by-step solution

1. 1

   Write the system in matrix form.

   $\begin{pmatrix}5.20 & 5.75\\ 1 & 1\end{pmatrix}\begin{pmatrix}X\\ Y\end{pmatrix}=\begin{pmatrix}8947.2\\ 1595\end{pmatrix}$

   Cramer’s rule expresses each unknown as a ratio of determinants, which is convenient for a $2\times 2$ system because each determinant is a single cross-multiplication.

2. 2

   Compute the coefficient determinant.

   $D=\begin{vmatrix}5.20 & 5.75\\ 1 & 1\end{vmatrix}=5.20(1)-5.75(1)=-0.55$

   It is nonzero, so the system has exactly one solution. Notice $|D|$ is again the price gap.

3. 3

   Compute the two numerator determinants. Replace the $X$-column, then the $Y$-column, by the right-hand side:

   $D_X=\begin{vmatrix}8947.2 & 5.75\\ 1595 & 1\end{vmatrix}=8947.2-5.75(1595)=8947.2-9171.25=-224.05$

   $D_Y=\begin{vmatrix}5.20 & 8947.2\\ 1 & 1595\end{vmatrix}=5.20(1595)-8947.2=8294-8947.2=-653.2$

4. 4

   Divide to get the unknowns.

   $X=\frac{D_X}{D}=\frac{-224.05}{-0.55}=\frac{4481}{11}\approx 407.36,\qquad Y=\frac{D_Y}{D}=\frac{-653.2}{-0.55}=\frac{13064}{11}\approx 1187.64$

   Both are repeating decimals, so the fractional forms are the exact answers.

5. 5

   Verify in the original equations. Sum: $\tfrac{4481+13064}{11}=\tfrac{17545}{11}=1595\;\checkmark$. Value, cleared of decimals by $\times 100$: $\tfrac{520\cdot 4481+575\cdot 13064}{11}=\tfrac{2330120+7511800}{11}=\tfrac{9841920}{11}=894720$, which is $8947.2\times 100\;\checkmark$.