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The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
use the elimination method to find all solutions
(for this equation i kept getting an answer with roots however the program I am using doesnt give me the option to input a root symbol. )
![### Solving Systems of Equations Using the Elimination Method
To find all solutions of the given system of equations using the elimination method, consider the following system:
\[
\begin{cases}
x^2 - y^2 + 3 = 0 \\
2x^2 + y^2 - 4 = 0
\end{cases}
\]
#### Step-by-Step Solution:
1. **Write down the equations:**
\[
\begin{aligned}
& 1. \quad x^2 - y^2 + 3 = 0 \\
& 2. \quad 2x^2 + y^2 - 4 = 0
\end{aligned}
\]
2. **Eliminate \( y^2 \) by adding the two equations:**
\[
\begin{aligned}
& (x^2 - y^2 + 3) + (2x^2 + y^2 - 4) = 0 \\
& x^2 + 2x^2 + 3 - 4 = 0 \\
& 3x^2 - 1 = 0 \\
& 3x^2 = 1 \\
& x^2 = \frac{1}{3} \\
& x = \pm \sqrt{\frac{1}{3}}
\end{aligned}
\]
3. **Substitute \( x = \sqrt{\frac{1}{3}} \) and \( x = -\sqrt{\frac{1}{3}} \) back into one of the original equations to solve for \( y \).**
4. **For \( x = \sqrt{\frac{1}{3}} \):**
\[
\begin{aligned}
& (\sqrt{\frac{1}{3}})^2 - y^2 + 3 = 0 \\
& \frac{1}{3} - y^2 + 3 = 0 \\
& -y^2 + 3 + \frac{1}{3} = 0 \\
& -y^2 + \frac{10}{3} = 0 \\
& -y^2 = -\frac{10}{3} \\
& y^2 = \frac{10}{3} \\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a7c8b7c-6421-40e7-b33a-5c89a0da3579%2F09b599a3-ef65-4f1e-800b-2894194bc48e%2Fjst2g.png&w=3840&q=75)
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