For what value of (a) the system of equations x ^ 2-y = 0 y-2x + a = 0 has a unique solution.

1. Let’s solve the system of two equations by the substitution method:

{x ^ 2 – y = 0;
{y – 2x + a = 0;

{y = x ^ 2;
{y – 2x + a = 0;

{y = x ^ 2;
{x ^ 2 – 2x + a = 0.

2. Let’s select the complete square of the binomial:

{y = x ^ 2;
{x ^ 2 – 2x + 1 – 1 + a = 0;

{y = x ^ 2;
{(x – 1) ^ 2 = 1 – a. (one)

3. Equation (1) has a unique solution provided:

1 – a = 0;

a = 1.

4. Substitute the value a = 1 and solve the system:

{y = x ^ 2;
{(x – 1) ^ 2 = 0;

{y = x ^ 2;
{x – 1 = 0;

{y = 1;
{x = 1.

Answer: the system of equations has a unique solution (1; 1) for a = 1.