Solve the System of Equations y=x-2 y=x^2-2

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:

1. Combining Equations: When two expressions are set equal to the same variable, they can be set equal to each other.

2. Solving Quadratic Equations: The process of factoring and using the zero product property to find solutions to quadratic equations.

3. Zero Product Property: If the product of two factors is zero, at least one of the factors must be zero.

4. Substitution Method: Substituting the found values of $x$ back into the original equations to solve for $y$.

5. Solution Set: The set of all ordered pairs that satisfy the system of equations.

6. LaTeX Formatting: The use of LaTeX to properly format mathematical expressions for clarity and precision.