\begin{aligned} x - y &= 3 \\ y &= x^2 - 7x + 12 \end{aligned} If (x, y) is a solution to … — SAT

\begin{aligned} x - y &= 3 \\ y &= x^2 - 7x + 12 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $xy$ ?

Cevap: 10

Cevap

By substituting the linear equation

y = x - 3

into the quadratic equation, we obtain a single quadratic equation

x^2 - 8x + 15 = 0

. Factoring this equation yields the solutions

x = 3

and

x = 5

. The corresponding

y

-coordinates are

y = 0

and

y = 2

, respectively. Since the problem specifies that

y > 0

, we choose the solution

(5, 2)

. The product of the coordinates is

5 \cdot 2 = 10

Adım Adım Çözüm

Express

y

in terms of

x

from the linear equation.

y = x - 3

This allows for substitution into the second equation.

Substitute

y = x - 3

into the quadratic equation.

x - 3 = x^2 - 7x + 12

To create a single quadratic equation in terms of

x

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

x^2 - 8x + 15 = 0

Subtracting

x

and adding

3

to both sides enables factoring.

Factor the quadratic expression.

(x - 3)(x - 5) = 0

To find the roots of the equation, which are

x = 3

and

x = 5

Find the corresponding

y

-values for each

x

-value.

For

x = 3

y = 3 - 3 = 0

. For

x = 5

y = 5 - 3 = 2

To obtain the complete coordinate pairs of the intersection points.

Apply the constraint

y > 0

to select the correct solution pair.

The solution

(5, 2)

is selected since its

y

-coordinate is greater than

0

The other solution,

(3, 0)

, has

y = 0

, which does not satisfy the constraint

y > 0

Calculate the value of

xy

for the chosen solution.

xy = 5 \cdot 2 = 10

To find the final requested value.

Anahtar Kavram

Solving a nonlinear system of equations containing a linear equation and a quadratic equation by substitution.

Tahmini Süre:1m 30s

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