Question

Решить систему неравенств методом подстановки

Answer 1

$\left \{ {{y=3x-10} \atop {x^2-y^2=20-xy}} \right.$
x²-(3x-10)²=20-x(3x-10);
x²-(9x²-60x+100)=20-3x²+10x;
-5x²+50x-120=0
x²-10x+24=0
По теореме Виета:
x₁+x₂=10; x₁*x₂=24
x₁=6; x₂=4
y=3x-10
y₁=18-10=8
y₂=12-10=2
Ответ: (6;8)∪(4;2)

Answer 2

Solving a System of Inequalities by Substitution

To solve a system of inequalities using the method of substitution, we need to find the values of the variables that satisfy both inequalities. Let's consider the system of inequalities you provided:

1) \(x^2 - 5x + 6 > 0\) 2) \(x - y = 5\)

To solve this system, we can start by solving the second inequality for one variable and substituting it into the first inequality.

From the second inequality, we have \(x = y + 5\). We can substitute this expression for \(x\) in the first inequality:

\((y + 5)^2 - 5(y + 5) + 6 > 0\)

Simplifying this inequality, we get:

\(y^2 + 10y + 25 - 5y - 25 + 6 > 0\)

Combining like terms, we have:

\(y^2 + 5y + 6 > 0\)

Now, let's solve this quadratic inequality. We can factor it as:

\((y + 2)(y + 3) > 0\)

To determine the sign of the inequality, we can analyze the signs of the factors:

- When \(y < -3\), both factors are negative, so the inequality is satisfied. - When \(-3 < y < -2\), the factor \((y + 2)\) is negative, but \((y + 3)\) is positive, so the inequality is not satisfied. - When \(y > -2\), both factors are positive, so the inequality is satisfied.

Therefore, the solution to the quadratic inequality is \(y < -3\) or \(y > -2\).

Now, let's substitute these values of \(y\) back into the second inequality to find the corresponding values of \(x\):

- For \(y < -3\), we have \(x = y + 5 < -3 + 5 = 2\). - For \(y > -2\), we have \(x = y + 5 > -2 + 5 = 3\).

Therefore, the solution to the system of inequalities is:

- When \(y < -3\), the solution is \(x < 2\). - When \(y > -2\), the solution is \(x > 3\).

In summary, the solution to the system of inequalities is \(x < 2\) or \(x > 3\) when \(y < -3\) or \(y > -2\).

Please note that the solution may vary depending on the specific values of the inequalities provided.