Linear equationsSee approximately page 50.Linear equations are multiple mathematical equations with multiple unknowns. Form of “y = mx+b”. Can be solved four ways:
- Graphically
- Substitution
- Elimination (or Addition)
- Matrices
Graphically solving linear equations means graphing two (or more) lines, and determining where they intersect, if they do. If the lines intersect at a point, that point is the solution. If there is more than one point of intersections, there are multiple solutions. A set of equations with no intersection (parallel lines) is inconsistent. A set of equations which all describe the same line has infinite solutions, is consistent, and is _______.
Solving linear equations via substitution means comparing them without graphing. To solve, first solve one equation of the set for a single variable. That is, find the value of one variable as a function of the other variable. Next, substitute that function for the the solved variable in the second equation. For a two-variable system, this should create an equation with a single unknown, and be solvable. For systems with more than two variables, those steps may need to be repeated. When you have an equation with a single unknown, solve that equation for the value of that variable. Substitute that value in the other equation(s) to solve for the remaining variables, repeating as needed. If all variables in an equation cancel themselves out leaving a true statement, the equations are consistent and there are infinitely many solutions. If all variables in a statement cancel themselves out leaving a false statement, the equations are inconsistent and there is no solution.
Solving systems of linear equations via elimination is also known as eliminating via addition, because ou add the two equations to remove variable. Select two equations from the system and eliminate a variable from them. (Page 55) To eliminate a variable, multiply all elements of one equation by the value needed to make one variable in that equation equivalent in size but opposite in value to the same variable in the oher equation. For instance, 5x and -5x are equivalent in size but opposite in value. Adding them cancels results in a sum of zero, which is the goal. After applying the multiplication, add all the factors of one equation to the matching factors of the other equation. Values of x are addes to other values of x, values of y are added to other values of y, constants are added to constants, and so forth. The result should be a single equation wih a single variable, which is therefore solvable. After solving for the remaining variable, substitute it into one of the original equations and solve for the value of the second variable.
Next week:Â Matrices