Glossary
Elimination Method
An algebraic technique for solving systems of equations by adding or subtracting equations to cancel out one variable.
Example:
To solve 3x + 2y = 12 and 2x - y = 5, you could use the elimination method by multiplying the second equation by 2 to make the 'y' coefficients opposites, then adding the equations.
Graphing Method
A visual technique for solving systems of equations by plotting each line on a coordinate plane and identifying their intersection point.
Example:
To find where the cost of two different phone plans becomes equal, you could use the graphing method to plot their cost functions and see where they cross.
Infinitely Many Solutions
A type of system where the equations represent the exact same line, meaning every point on the line is a solution.
Example:
If you have two identical recipes for cookies, any batch size you make will satisfy both, illustrating a system with infinitely many solutions.
Interpreting Solutions
The process of explaining what the numerical solution of a system means in the context of a given real-world problem.
Example:
If your solution is (5, 2) for a problem about fruit, interpreting solutions means stating, 'The cost of an apple is 2.'
No Solution
A type of system where the lines are parallel and never intersect.
Example:
Trying to find a time when two trains on parallel tracks will collide is an example of a system with no solution.
One Solution
A type of system where the lines intersect at exactly one unique point.
Example:
If two friends start walking from different points towards each other, the moment they meet represents a system with one solution.
Solution (of a system)
An ordered pair (x, y) that satisfies all equations within a given system simultaneously.
Example:
For the system y = x + 1 and y = -x + 5, the solution is (2, 3) because plugging in x=2 and y=3 works for both equations.
Substitution Method
An algebraic technique for solving systems of equations by solving one equation for a variable and then plugging that expression into the other equation.
Example:
To solve 2x + y = 7 and x - y = -1, you could use the substitution method by rewriting the second equation as x = y - 1 and plugging that into the first.
Systems of Linear Equations
A set of two or more linear equations that share the same variables, used to find values that satisfy all equations simultaneously.
Example:
When trying to find the price of both a taco and a burrito, you might set up a system of linear equations to represent the total cost of different meal combinations.
Variables (in real-world problems)
Symbols (like x or y) used to represent unknown quantities or real-world entities within an equation or system.
Example:
In a problem about selling tickets, 'x' might represent the number of adult tickets and 'y' the number of child tickets, acting as your variables.