Glossary
Algebraic Solution (Mixed Systems)
The primary method for solving linear and quadratic systems, involving substituting the linear equation into the quadratic equation to form a single quadratic equation, which is then solved for x.
Example:
To find where y = x + 1 intersects y = x^2 - 3x + 4, you would use the algebraic solution by setting x + 1 = x^2 - 3x + 4.
Discriminant
The part of the quadratic formula under the square root, b^2 - 4ac, which determines the number of real solutions a quadratic equation has.
Example:
If the discriminant of a quadratic equation is negative, you know immediately that the line and parabola in a mixed system do not intersect.
Elimination Method
An algebraic technique to solve systems of equations by adding or subtracting the equations to eliminate one of the variables. This is often done after multiplying one or both equations by a constant.
Example:
To solve 2x + y = 7 and x - y = 2, you could use the elimination method by adding the two equations together to cancel out the y terms.
Graphing Method
A visual technique to solve systems of equations by plotting each equation on a coordinate plane. The solution is the point(s) where the graphs intersect.
Example:
When trying to visualize the break-even point for a business, you might use the graphing method to see where the cost and revenue lines cross.
Infinitely Many Solutions (Linear Systems)
A type of solution for a system of linear equations where the equations represent the exact same line. Every point on the line is a solution, meaning there are countless (x, y) pairs that satisfy both equations.
Example:
If you write the equation of a line as y = 2x + 3 and then rewrite it as 2y = 4x + 6, these two equations have infinitely many solutions because they are identical.
Linear and Quadratic Systems
A system of equations that includes one linear equation (a straight line) and one quadratic equation (a parabola). These systems can have zero, one, or two solutions.
Example:
Finding the points where a thrown ball's parabolic path intersects a laser beam's straight path involves solving a linear and quadratic system.
No Solution (Linear Systems)
A type of solution for a system of linear equations where the lines are parallel and never intersect. This indicates that there is no (x, y) pair that can satisfy both equations simultaneously.
Example:
Two train tracks running perfectly parallel to each other illustrate a no solution system, as they will never meet.
One Solution (Linear Systems)
A type of solution for a system of linear equations where the lines intersect at exactly one unique point. This point represents the single (x, y) pair that satisfies both equations.
Example:
The intersection of a road going north-south and a road going east-west represents a one solution scenario for their paths.
Slope
A measure of the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It is crucial for determining if lines are parallel or perpendicular.
Example:
A ski slope with a slope of -1 indicates a steep descent, while a flat path has a slope of 0.
Substitution Method
An algebraic technique to solve systems of equations by solving one equation for a variable and then plugging that expression into the other equation. This reduces the system to a single equation with one variable.
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.
System of Linear Equations
A set of two or more linear equations that share the same variables. The solution is the point(s) where all lines intersect.
Example:
To find the optimal price for a new gadget, you might set up a system of linear equations to model supply and demand.
Tangent
A line that touches a curve (like a parabola) at exactly one point without crossing it. In linear and quadratic systems, a tangent line indicates exactly one solution.
Example:
A skateboarder briefly touching the edge of a half-pipe at its peak before continuing their descent is tangent to the curve of the ramp.
Y-intercept
The point where a line crosses the y-axis on a coordinate plane. It is the value of y when x is equal to zero.
Example:
In a graph showing a car's distance from home over time, the y-intercept would represent the car's starting distance from home.