Mark44 said:Think about what the augmented matrix represents: Ax = c, where A is your 3 x 3 matrix, x is the vector <x, y, z>, and c is the vector of constants, <a, b, c>.
The bottom row means 0x + 0y + 0z = b - a - c. Regardless of the values of x, y, and z, what is the only value that the right side could have to make an equation that had a solution?
Mark44 said:Right. So the system of equations is consistent provided that b - c - a = 0. If two variables are specified, the third can be found. That means that there are two free variables, and therefore, the set of points in R3 for which the system of equations is consistent is a plane in space.
A linear equation in terms of a, b, and c is an equation in the form of ax + by + c = 0, where a, b, and c are constants and x and y are variables. It represents a straight line on a graph and can be used to solve for the values of x and y.
To solve for the values of a, b, and c in a linear equation, you can use a system of equations. Set up two equations with two unknowns using the given values and then solve for the unknowns using algebraic methods such as substitution or elimination.
The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis). This form is useful for graphing and can be derived from the standard form ax + by + c = 0.
Yes, a linear equation can have more than two variables. In fact, the general form of a linear equation is ax1 + bx2 + ... + zxn + c = 0, where n is the number of variables. However, the most common forms of linear equations have only two variables.
Linear algebra is used in many real-life applications, including computer graphics, data analysis, engineering, and economics. It is also used in physics and chemistry to model and solve systems of equations. Additionally, linear algebra is the foundation for more advanced mathematical concepts such as calculus and differential equations.