1. **Problem statement:** We have two linear equations: $$a \cdot x + b \cdot y = c$$ and $$x + y = 1,$$ with the additional condition $$\frac{1}{a} = \frac{3}{4} \cdot b \cdot (x - 1)^2,$$ and the domain constraint $$0 \leq x \leq 0.5.$$ 2. **Step 1: Express $y$ from the second equation.** From $$x + y = 1,$$ solve for $y$: $$y = 1 - x.$$ 3. **Step 2: Substitute $y$ into the first equation.** Replace $y$ with $1 - x$ in $$a \cdot x + b \cdot y = c,$$ we get: $$a \cdot x + b \cdot (1 - x) = c.$$ 4. **Step 3: Simplify the substituted equation.** Distribute $b$: $$a \cdot x + b - b \cdot x = c.$$ Group like terms: $$(a - b) x + b = c.$$ 5. **Step 4: Solve for $x$.** Isolate $x$: $$(a - b) x = c - b,$$ so $$x = \frac{c - b}{a - b}.$$ 6. **Step 5: Use the condition involving $1/a$.** Given: $$\frac{1}{a} = \frac{3}{4} \cdot b \cdot (x - 1)^2.$$ Rewrite to express $a$: $$a = \frac{1}{\frac{3}{4} b (x - 1)^2} = \frac{4}{3 b (x - 1)^2}.$$ 7. **Step 6: Substitute $a$ into the $x$ solution from Step 4.** Compute: $$x = \frac{c - b}{a - b} = \frac{c - b}{\frac{4}{3 b (x - 1)^2} - b}.$$ This is a complicated implicit equation for $x$. 8. **Step 7: Apply the domain constraint:** $$0 \leq x \leq 0.5.$$ We look for $x$ values in this range that satisfy the implicit equation. 9. **Step 8: Further remarks.** Because of the implicit nature, one would typically use numerical methods or graphing to find $x$ given $b, c$. 10. **Final summary:** - From the second equation: $y = 1 - x$. - $a$ is related to $b$ and $x$ by: $$a = \frac{4}{3 b (x - 1)^2}.$$ - $x$ satisfies: $$x = \frac{c - b}{a - b},$$ with $a$ as above. - The constraint $0 \leq x \leq 0.5$ limits acceptable solutions. Therefore, the solution depends on known values of $b$ and $c$ and requires substitution or numeric evaluation for $x$ and then $y$.