12/14 12. Systems of Equations and Inequalities Free Online FlashCards

What is a system of equations?

A system is a collection of two or more equations considered simultaneously.

How do two lines classify a linear system?

One intersection gives a consistent independent system; parallel distinct lines give no solution and an inconsistent system; coincident lines give infinitely many solutions and a dependent system.

What is the substitution method?

Substitution solves one equation for a variable, replaces that variable in the other equation, solves, and then substitutes back to find the remaining variable.

What do contradiction and identity reveal?

A contradiction such as $0=5$ means the system has no solution. An identity such as $0=0$ means the system has infinitely many solutions.

How does the elimination method work?

Elimination combines equations, often after multiplying one or both by constants, so that one variable cancels. The remaining variable is then solved and substituted back.

How are systems with three variables reduced?

For a three-variable system, repeated elimination reduces the equations to systems with fewer variables. A free variable can remain in a dependent system, producing infinitely many solutions.

What does an augmented matrix contain?

An augmented matrix places the constants beside the coefficient matrix, separated by a vertical bar. For example, the system coefficients and constants form one compact array.

Which row operations preserve solutions?

The elementary row operations are swapping rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another. Each preserves the solution set.

When is a two-by-two matrix invertible?

For a two-by-two matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, the determinant is $ad-bc$. The matrix is invertible when this value is nonzero.

How do inequality symbols determine boundary lines?

Use a solid boundary for $\le$ or $\ge$, because boundary points are included. Use a dashed boundary for $<$ or $>$, because they are excluded.

How does a test point choose shading?

A test point determines which side of a boundary satisfies the inequality. Substitute its coordinates and shade the side where the resulting statement is true.

What equations model the ticket-sales example?

For the ticket model, $a+s=250$ represents the total tickets and $12a+8s=2600$ represents total revenue. Solving gives 150 adult and 100 student tickets.