Systems of equations two prices for a test

What is a system of equations?

A collection of two or more equations with the same variables.

Example scenario: buying apples and oranges.

You buy apples for $\displaystyle 2 each and oranges for$3 each.

Solve: 2x + 3y = 12

x = 3, y = 2. This represents quantities of apples and oranges.

True or false: Systems can have no solution.

True, because two lines can be parallel.

What does (x, y) represent in a graph?

The coordinates where the lines of equations intersect.

Difference between consistent and inconsistent systems.

Consistent systems have at least one solution; inconsistent has none.

Fill in the blank: A solution to a system is where ____.

the equations intersect on a graph.

What is the substitution method?

A method to solve systems by expressing one variable in terms of another.

True or false: All systems of equations are linear.

False, some can be nonlinear.

What do you need to solve a system?

Two equations with the same variables.

Quiz: If apples cost $\displaystyle 2 and oranges$3, what’s 4 apples + 2 oranges?

$\displaystyle 2(4) +$3(2) = $14.

What is the elimination method?

A technique where you add or subtract equations to eliminate a variable.

Scenario: You have $\displaystyle 20 and buy pencils for$1 and erasers for $2. Write an equation.

Let x = pencils, y = erasers: $\displaystyle 1x +$2y = $20.

Example of a dependent system.

2x + 4y = 8 and x + 2y = 4 are the same line.

What does the slope represent in a system?

The rate of change of one variable in relation to another.

Fill in the blank: To graph a system, plot the ____ first.

individual equations.

Compare: graphing vs substitution methods.

Graphing visually shows solutions; substitution uses algebra to find them.

What can you find at the intersection of two lines?

The solution(s) to the system of equations.