Glossary
Creating Accurate Equations
The process of translating the relationships described in a word problem into correct mathematical expressions, ensuring the number of equations matches the number of unknowns.
Example:
If 'the sum of two numbers is 15' and 'their difference is 3', you'd be creating accurate equations like x + y = 15 and x - y = 3.
Defining Variables
The crucial first step in setting up word problems, where you clearly state what each variable (e.g., x, y) represents in the context of the problem.
Example:
Before writing equations, you should always define variables like 'let x = the number of hours worked' and 'let y = the hourly wage'.
Elimination Method
A technique for solving systems of linear equations by adding or subtracting the equations to cancel out one of the variables, allowing you to solve for the remaining variable.
Example:
To solve 2x + y = 7 and x - y = 2, you could use the elimination method by adding the two equations together to remove 'y'.
Graphing Method
A visual technique for solving systems of linear equations by plotting both equations on a coordinate plane and finding the point where their lines intersect.
Example:
While less common on the digital SAT, you could use the graphing method to find the solution to y = x + 2 and y = -2x + 5 by seeing where their lines cross.
Infinite Solutions (Coincident Lines)
A special case where the two equations in a system represent the exact same line, meaning every point on the line is a solution and there are countless combinations that satisfy both equations.
Example:
If you have x + y = 5 and 2x + 2y = 10, these are essentially the same line, leading to infinite solutions because they are coincident lines.
Interpreting Solutions
Understanding what the numerical values obtained from solving a system of equations actually mean in the context of the original word problem.
Example:
If your solution is x = 5 and y = 10, interpreting solutions means realizing that 'x' might be the number of small pizzas and 'y' the number of large pizzas.
No Solution (Parallel Lines)
A special case in systems of linear equations where the lines represented by the equations are parallel and never intersect, indicating that no values satisfy both equations simultaneously.
Example:
If you try to solve y = 2x + 3 and y = 2x - 1, you'll find no solution because the lines are parallel and will never cross.
Realistic Constraints
Considering whether the numerical solution makes practical sense within the context of the real-world problem, such as not having negative quantities or fractional people.
Example:
If you're solving for the number of students, a solution of -5 or 3.75 would violate realistic constraints.
Substitution Method
A technique for solving systems of linear equations by solving one equation for a variable and then plugging that expression into the other equation.
Example:
If you have y = 2x + 1 and 3x + y = 11, you can use the substitution method by replacing 'y' in the second equation with '2x + 1'.
Systems of Linear Equations
A set of two or more linear equations that share the same variables, representing a puzzle where you need to find the values of those variables that satisfy all equations simultaneously.
Example:
To find the cost of apples and bananas, you might set up a system like 2a + 3b = 7 and 4a + b = 9.
Units
The standard measurements (e.g., dollars, hours, miles per hour) that should be included with numerical answers to provide context and clarity in real-world problems.
Example:
When calculating the cost, always include units like '15.50' instead of just '15.50'.
Unknown Quantities (Variables)
The values that are not given in a problem and need to be determined, typically represented by letters like 'x' and 'y' in linear equations.
Example:
In a problem about ticket sales, the number of adult tickets and child tickets would be your unknown quantities.
Verifying Solutions
The process of plugging the calculated values of the variables back into the original equations to ensure they satisfy all conditions and make the equations true.
Example:
After finding x=3 and y=4 for x+y=7 and 2x-y=2, you would verify solutions by checking if 3+4=7 and 2(3)-4=2.