Solve the System of Equations x^2-y=1 2x^2+y^2=17
The given problem is asking to find the values of the variables x and y that satisfy both of the equations in the system simultaneously. The system includes two equations: the first equation is a parabola (x^2 - y = 1), and the second equation represents an ellipse (2x^2 + y^2 = 17). The solution involves finding the point(s) where these two curves intersect, which equates to finding the pairs of (x, y) that make both equations true when plugged into them.
$x^{2} - y = 1$$2 x^{2} + y^{2} = 17$
Answer
Solution:
Step 1: Isolate $y$ in the first equation $x^2 - y = 1$.
Subtract $x^2$ from both sides: $-y = 1 - x^2$.
Multiply by $-1$: $y = x^2 - 1$.
Step 2: Substitute $y$ with $x^2 - 1$ in the second equation $2x^2 + y^2 = 17$.
Replace $y$: $2x^2 + (x^2 - 1)^2 = 17$.
Expand and simplify: $2x^2 + (x^4 - 2x^2 + 1) = 17$.
Combine like terms: $x^4 + 1 = 17$.
Step 3: Solve for $x$ in the equation $x^4 + 1 = 17$.
Subtract 1 from both sides: $x^4 = 16$.
Take the fourth root: $x = \pm 2$.
Step 4: Find $y$ for each $x$ value.
For $x = 2$: $y = 2^2 - 1 = 3$.
For $x = -2$: $y = (-2)^2 - 1 = 3$.
Step 5: The solution set is the pair of points where the equations intersect.
- The solutions are $(2, 3)$ and $(-2, 3)$.
Knowledge Notes:
To solve a system of equations, one can use substitution or elimination methods. In this case, we used substitution, which involves the following steps:
Isolate one variable in one of the equations.
Substitute the expression for the isolated variable into the other equation.
Solve the resulting equation for the remaining variable.
Substitute the found value(s) back into any of the original equations to find the corresponding value(s) of the other variable.
Verify the solution by plugging the values back into the original equations.
In the context of this problem, we dealt with a system of nonlinear equations involving squares of variables. When solving such systems, it's often necessary to manipulate the equations to isolate terms and simplify expressions. This can include expanding binomials, combining like terms, and taking roots of numbers.
The FOIL method (First, Outer, Inner, Last) is a technique used to expand binomials. When you have a binomial squared, such as $(a + b)^2$, it expands to $a^2 + 2ab + b^2$.
The power rule for exponents states that $a^m \cdot a^n = a^{m+n}$. This rule is used to combine like terms when variables are raised to powers.
Taking roots is the inverse operation of exponentiation. For example, taking the fourth root of $x^4$ gives us $x$, and the fourth root of $16$ is $2$ because $2^4 = 16$.
The final solution to a system of equations is the set of all ordered pairs that satisfy all equations in the system. In this case, the solutions are presented as points in the Cartesian coordinate system.
