Advanced Math / Systems Equations Two Variables
Systems of Equations in Two Variables
Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.
Least you need to know
- A solution to a system is an ordered pair that satisfies every equation in the system.
- Substitution is often efficient when one equation is already solved for one variable.
- The number of real solutions can often be determined by the discriminant of the resulting quadratic.
Key notation
- $(x,y)$ — An ordered pair that may solve all equations in a system
- $x+y=s$ and $xy=p$ — A system that can often be connected to the roots of a quadratic
Worked example
- Consider $y=x+2$ and $y=x^2-4$.
- Substitute $x+2$ for $y$: $x+2=x^2-4$.
- Rearrange: $x^2-x-6=0$.
- Factor: $(x-3)(x+2)=0$.
- Thus, $x=3$ or $x=-2$.
Common mistakes
- A system may have two, one, or no real solutions.
- If solving by substitution produces a quadratic, both roots may correspond to solutions.
- Check whether the question asks for an $x$-coordinate, a $y$-coordinate, an ordered pair, or an expression involving both variables.
How to recognize it
- The system includes at least one nonlinear equation such as a quadratic, circle-like equation, product equation, or rational relation.
- The problem asks for an intersection coordinate, a sum or product of coordinates, or the number of solutions.
- One equation may be designed for direct substitution into the other.
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