Solve the system: X * 4.50 + Y * 4.95 = 2368.15 and X + Y = 490.5.

Step-by-step solution

1. 1

   Solve the simple equation for one variable. The total equation rearranges with no arithmetic at all:

   $X=490.5-Y$

   Substitution is the natural choice whenever one equation already has a variable with coefficient $1$.

2. 2

   Substitute into the value equation.

   $4.50(490.5-Y)+4.95Y=2368.15$

   $2207.25-4.50Y+4.95Y=2368.15$

3. 3

   Collect and solve for $Y$. The $Y$ coefficient is the price gap $4.95-4.50=0.45$:

   $0.45Y=2368.15-2207.25=160.90\;\Longrightarrow\;Y=\frac{160.90}{0.45}=\frac{16090}{45}=\frac{3218}{9}\approx 357.56$

4. 4

   Recover $X$ from the substitution.

   $X=490.5-\frac{3218}{9}=\frac{4414.5-3218}{9}=\frac{2393}{18}\approx 132.94$

   Keeping $\tfrac{2393}{18}$ rather than a rounded decimal matters if the answer feeds into further calculation.

5. 5

   Verify both equations. Sum: $\tfrac{2393}{18}+\tfrac{3218}{9}=\tfrac{2393+6436}{18}=\tfrac{8829}{18}=490.5\;\checkmark$. Value: $4.50\cdot\tfrac{2393}{18}+4.95\cdot\tfrac{3218}{9}=\tfrac{10768.5}{18}+\tfrac{15929.1}{9}=598.25+1769.90=2368.15\;\checkmark$.