Solve the System of Equations y=x-2 y=x^2-2
The problem requires finding the values of x and y that simultaneously satisfy two different equations: a linear equation, y = x - 2, and a quadratic equation, y = x^2 - 2. You are being asked to determine the points at which the graph of the straight line intersects with the graph of the parabola, which involves solving a system of equations.
$y = x - 2$$y = x^{2} - 2$
Answer
Solution:
Step:1
Combine the two equations by setting them equal to each other since they both equal $y$. Thus, $x - 2 = x^{2} - 2$.
Step:2
Isolate $x$ in the equation $x - 2 = x^{2} - 2$.
Step:2.1
Move $x^{2}$ to the left side by subtracting it from both sides: $x - x^{2} - 2 = -2$.
Step:2.2
Cancel out the $-2$ on both sides by adding $2$: $x - x^{2} = 0$.
Step:2.3
Simplify the equation by combining like terms.
Step:2.3.1
Combine $-2$ and $2$ to get $0$: $x - x^{2} = 0$.
Step:2.3.2
The equation is already simplified to $x - x^{2} = 0$.
Step:2.4
Extract the common factor $x$ from the terms.
Step:2.4.1
Take $x$ out of $x$: $x(1) - x^{2} = 0$.
Step:2.4.2
Take $x$ out of $-x^{2}$: $x(1 - x) = 0$.
Step:2.4.3
Write the factored form: $x(1 - x) = 0$.
Step:2.5
Apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must be zero.
Step:2.6
Set $x$ equal to zero: $x = 0$.
Step:2.7
Solve for $x$ when $1 - x = 0$.
Step:2.7.1
Set $1 - x$ equal to zero: $1 - x = 0$.
Step:2.7.2
Isolate $x$: $x = 1$.
Step:3
Find the corresponding $y$ values when $x = 0$.
Step:3.1
Plug $x = 0$ into the second equation: $y = 0^{2} - 2$.
Step:3.2
Calculate $y$: $y = -2$.
Step:4
Find the corresponding $y$ values when $x = 1$.
Step:4.1
Plug $x = 1$ into the second equation: $y = 1^{2} - 2$.
Step:4.2
Calculate $y$: $y = -1$.
Step:5
Combine the $x$ and $y$ values to form the solution set to the system of equations.
Step:6
Express the solution in different forms.
Point Form:
$(0, -2), (1, -1)$
Equation Form:
For $x = 0$, $y = -2$.
For $x = 1$, $y = -1$.
Knowledge Notes:
The problem involves solving a system of equations where both equations are set equal to $y$. The steps include manipulating the equations to isolate $x$, applying the zero product property, and then finding the corresponding $y$ values for each solution of $x$.
Relevant knowledge points include:
Combining Equations: When two expressions are set equal to the same variable, they can be set equal to each other.
Solving Quadratic Equations: The process of factoring and using the zero product property to find solutions to quadratic equations.
Zero Product Property: If the product of two factors is zero, at least one of the factors must be zero.
Substitution Method: Substituting the found values of $x$ back into the original equations to solve for $y$.
Solution Set: The set of all ordered pairs that satisfy the system of equations.
LaTeX Formatting: The use of LaTeX to properly format mathematical expressions for clarity and precision.
