SAT Math 3: Systems of Linear Equations

System of linear equations

A collection of two or more linear equations involving the same variables, with all equations required to be satisfied simultaneously.

What does it mean for an ordered tuple to solve a system of linear equations?

The tuple assigns values to all variables so that substitution makes every equation in the system true.

What is the solution set of a linear system?

It is the set of all assignments of variable values that satisfy every equation. A system can have no solution, exactly one solution, or infinitely many solutions.

Geometric meaning of a linear equation in two variables

Each equation represents a line in the coordinate plane. The solution set of the system is the intersection of the lines.

How does graphing a system of two linear equations reveal its solution?

Graph both lines. Any intersection point gives the values of $x$ and $y$ that satisfy both equations.

How can the number of solutions of a graphed system be identified?

Intersecting lines have one solution, parallel distinct lines have no solution, and coincident lines have infinitely many solutions.

How can a system have infinitely many solutions?

The equations must represent the same line, meaning one equation is equivalent to a multiple of the other. Every point on that line satisfies both equations.

How can a system have no solution?

The equations can represent distinct parallel lines. Algebraically, solving the system produces a contradiction such as $0=5$.

What is the elimination-of-variables method?

Add or subtract equations, possibly after multiplying one or both equations by constants, to eliminate a variable. Solve the resulting one-variable equation, then substitute back to find the other variable.

What is the substitution method for solving a system?

Solve one equation for one variable, substitute that expression into the other equation, solve the resulting one-variable equation, and substitute back to find the remaining variable.

When is substitution especially convenient?

Substitution is especially convenient when one equation already has a variable isolated or can be isolated with little algebra.

When is elimination especially convenient?

Elimination is especially convenient when adding or subtracting the equations, perhaps after scaling one equation, causes a variable's coefficients to cancel.

How can a solution to a system be checked?

Substitute both proposed variable values into each original equation. The values are a solution only if every original equation is true.

What are equivalent systems of linear equations?

Two systems are equivalent when they have exactly the same solution set. Reversible algebraic transformations, such as adding a multiple of one equation to another, produce an equivalent system.

How can dependent equations be recognized in a two-equation system?

If one equation is a constant multiple of the other, the equations are dependent and represent the same line, so the system has infinitely many solutions.

What does an inconsistent system mean?

An inconsistent system has no solution because no assignment of values satisfies all of its equations.

How do you translate a real-world situation into a system of linear equations?

Define variables for the unknown quantities, write one linear equation for each independent condition in the situation, and solve the resulting system.

How should a solution to a word problem involving a system be interpreted?

Use the variable definitions to translate the algebraic solution back into the quantities in the situation, include appropriate units, and check that the values satisfy the original conditions.