Solving Linear-Nonlinear Systems 23-24

Front

What is a linear equation?

Back

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What is a nonlinear equation?

Back

A nonlinear equation is an equation that does not form a straight line when graphed. It can include variables raised to a power greater than one, such as quadratic equations (e.g., y = x² + 3x - 5).

Front

How do you solve a system of linear and nonlinear equations?

Back

To solve a system of linear and nonlinear equations, you can substitute the linear equation into the nonlinear equation or use graphing to find the points of intersection.

Front

What is the quadratic formula?

Back

The quadratic formula is used to find the solutions of a quadratic equation in the form ax² + bx + c = 0. It is given by x = (-b ± √(b² - 4ac)) / (2a).

Front

What does it mean for a system of equations to have no solution?

Back

A system of equations has no solution when the equations represent parallel lines that never intersect.

Front

What does it mean for a system of equations to have one solution?

Back

A system of equations has one solution when the equations represent lines that intersect at exactly one point.

Front

What does it mean for a system of equations to have infinitely many solutions?

Back

A system of equations has infinitely many solutions when the equations represent the same line, meaning they overlap completely.

Front

How can you check if your solution to a system of equations is correct?

Back

You can check your solution by substituting the values back into the original equations to see if they satisfy both equations.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots of a quadratic equation: if it's positive, there are two real solutions; if it's zero, there is one real solution; if it's negative, there are no real solutions.

Front

What is the vertex of a parabola?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. It can be found using the formula (-b/2a, f(-b/2a)) for the quadratic equation y = ax² + bx + c.