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Solve the system of equations:
[tex]\[
\left\{\begin{array}{l}
y = -6x - 16 \\
y = x^2 - 7
\end{array}\right.
\][/tex]


Sagot :

Certainly! Let's solve the system of equations step-by-step.

We have the following system of equations:

[tex]\[ \left\{ \begin{array}{c} y = -6x - 16 \\ y = x^2 - 7 \end{array} \right. \][/tex]

To solve this system, we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations. Here is the step-by-step solution:

1. Set the equations equal to each other:

Since both equations are equal to [tex]\( y \)[/tex], we can set them equal to each other:

[tex]\[ -6x - 16 = x^2 - 7 \][/tex]

2. Rearrange the equation:

Bring all terms to one side of the equation to set it to zero:

[tex]\[ x^2 + 6x + 9 = 0 \][/tex]

3. Factor the quadratic equation:

Notice that [tex]\( x^2 + 6x + 9 \)[/tex] is a perfect square trinomial. We can factor it as follows:

[tex]\[ (x + 3)^2 = 0 \][/tex]

4. Solve for [tex]\( x \)[/tex]:

Set each factor equal to zero:

[tex]\[ x + 3 = 0 \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ x = -3 \][/tex]

5. Solve for [tex]\( y \)[/tex]:

Use the value of [tex]\( x \)[/tex] in either of the original equations to find [tex]\( y \)[/tex]. Let's use the first equation [tex]\( y = -6x - 16 \)[/tex]:

[tex]\[ y = -6(-3) - 16 \][/tex]

Simplify:

[tex]\[ y = 18 - 16 \][/tex]

Thus:

[tex]\[ y = 2 \][/tex]

6. Write the solution:

We have found that [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex]. Therefore, the solution to the system of equations is:

[tex]\[ (-3, 2) \][/tex]

So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex].