SAT systems of equations flashcards

What is a system of equations?

A set of two or more equations with the same variables, solved together.

Graphical method

Graph both equations and find their intersection point for the solution.

True or false: Systems can have no solutions.

True, because parallel lines represent equations that never intersect.

Substitution method

Solve one equation for a variable, then substitute into the other to find the solution.

Elimination method

Add or subtract equations to eliminate a variable, then solve for the others.

Fill in the blank: The intersection of lines represents __________.

The solution to the system of equations.

Difference between consistent and inconsistent systems

Consistent systems have at least one solution, while inconsistent systems have none.

What is a dependent system?

A system with infinitely many solutions, where equations are multiples of each other.

Example of a real-world application

Finding cost: If apples cost $\displaystyle 2 each and oranges$3, how to express total cost?

True or false: A single equation can represent a system.

False, a system requires multiple equations.

How to check solutions?

Substitute the solution into both original equations to verify.

Example of substitution

For $\displaystyle y = 2x + 3$ and $\displaystyle x + y = 10$, substitute $\displaystyle y$ into the second equation.

Standard form of a linear equation

$\displaystyle Ax + By = C$, where A, B, and C are constants.

What does it mean when two lines are parallel?

The system of equations is inconsistent; they have no solutions.

Example of elimination

For $\displaystyle 2x + 3y = 6$ and $\displaystyle 4x + 6y = 12$, multiply the first equation and subtract.

True or false: Systems can have exactly two solutions.

False, systems can have one, none, or infinitely many solutions.

Cost example setup

If $\displaystyle x$ is the number of shirts at $\displaystyle 10 each$ and $\displaystyle y$ the number of pants at $\displaystyle 20 each$, $\displaystyle 10x + 20y = Total$.

What is an independent system?

A system with exactly one solution; lines intersect at a single point.

Fill in the blank: Inconsistent systems have __________.

No solutions, as the lines do not intersect.

Finding intersection in equations

Solve equations simultaneously or graphically determine where they meet.

Cost of items example

If two products total $\displaystyle 50, and one is$30 more than the other, set up a system to find individual prices.

What is a solution to a system?

Values of variables that make all equations true simultaneously.