Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Instructions: Solve the following system of equations algebraically. If there are no real solutions, write "none" in both blanks. If there is only one solution, write "none" in the second blank.

[tex]\[
\left\{
\begin{array}{l}
y = -x - 10 \\
y = x^2 - 5x - 10
\end{array}
\right.
\][/tex]

( [tex]$\square$[/tex] , [tex]$\square$[/tex] )


Sagot :

Sure, let's solve the given system of equations step-by-step.

The system of equations is:
[tex]\[ \left\{\begin{array}{l} y = -x - 10 \\ y = x^2 - 5x - 10 \end{array}\right. \][/tex]

1. Step 1: Set the equations equal to each other

Since both equations are equal to [tex]\( y \)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[ -x - 10 = x^2 - 5x - 10 \][/tex]

2. Step 2: Move all terms to one side of the equation to set it to zero

Add [tex]\( x + 10 \)[/tex] to both sides:
[tex]\[ 0 = x^2 - 5x - 10 + x + 10 \][/tex]

3. Step 3: Combine like terms

Simplify the equation:
[tex]\[ 0 = x^2 - 4x \][/tex]

4. Step 4: Factor the quadratic equation

Factor out [tex]\( x \)[/tex]:
[tex]\[ 0 = x(x - 4) \][/tex]

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

Set each factor equal to zero and solve:
[tex]\[ x = 0 \quad \text{or} \quad x = 4 \][/tex]

6. Step 6: Substitute each [tex]\( x \)[/tex] value back into one of the original equations to find [tex]\( y \)[/tex]

Let's use the equation [tex]\( y = -x - 10 \)[/tex]:

For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -0 - 10 = -10 \][/tex]
So, one solution is [tex]\((0, -10)\)[/tex].

For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -4 - 10 = -14 \][/tex]
So, another solution is [tex]\((4, -14)\)[/tex].

7. Conclusion:

The solutions to the system of equations are:
[tex]\[ (0, -10) \quad \text{and} \quad (4, -14) \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.