Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve the given system of equations:
[tex]\[ \begin{array}{l} y = 2x \\ y = x^2 - 15 \end{array} \][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
1. Substitute [tex]\( y = 2x \)[/tex] from the first equation into the second equation [tex]\( y = x^2 - 15 \)[/tex]:
[tex]\[ 2x = x^2 - 15 \][/tex]
2. Rearrange this equation to form a quadratic equation:
[tex]\[ x^2 - 2x - 15 = 0 \][/tex]
3. Solve the quadratic equation by factoring:
To factor [tex]\( x^2 - 2x - 15 \)[/tex], we need two numbers that multiply to [tex]\(-15\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(-5\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[ (x - 5)(x + 3) = 0 \][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 5 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
[tex]\[ x = 5 \quad \text{or} \quad x = -3 \][/tex]
5. Find the corresponding [tex]\( y \)[/tex] values using [tex]\( y = 2x \)[/tex]:
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 2(5) = 10 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 2(-3) = -6 \][/tex]
6. List the solutions as ordered pairs:
- [tex]\( (x, y) = (5, 10) \)[/tex]
- [tex]\( (x, y) = (-3, -6) \)[/tex]
Therefore, the solutions to the system of equations are [tex]\( (5, 10) \)[/tex] and [tex]\( (-3, -6) \)[/tex].
The correct answer is:
C. [tex]\( (-3, -6) \)[/tex] and [tex]\( (5, 10) \)[/tex]
[tex]\[ \begin{array}{l} y = 2x \\ y = x^2 - 15 \end{array} \][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
1. Substitute [tex]\( y = 2x \)[/tex] from the first equation into the second equation [tex]\( y = x^2 - 15 \)[/tex]:
[tex]\[ 2x = x^2 - 15 \][/tex]
2. Rearrange this equation to form a quadratic equation:
[tex]\[ x^2 - 2x - 15 = 0 \][/tex]
3. Solve the quadratic equation by factoring:
To factor [tex]\( x^2 - 2x - 15 \)[/tex], we need two numbers that multiply to [tex]\(-15\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(-5\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[ (x - 5)(x + 3) = 0 \][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 5 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
[tex]\[ x = 5 \quad \text{or} \quad x = -3 \][/tex]
5. Find the corresponding [tex]\( y \)[/tex] values using [tex]\( y = 2x \)[/tex]:
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 2(5) = 10 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 2(-3) = -6 \][/tex]
6. List the solutions as ordered pairs:
- [tex]\( (x, y) = (5, 10) \)[/tex]
- [tex]\( (x, y) = (-3, -6) \)[/tex]
Therefore, the solutions to the system of equations are [tex]\( (5, 10) \)[/tex] and [tex]\( (-3, -6) \)[/tex].
The correct answer is:
C. [tex]\( (-3, -6) \)[/tex] and [tex]\( (5, 10) \)[/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.