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Systems of Linear Equations and Inequalities (Chapter 3)
Graphing Systems of Equations (3.1) system of equations = two or more equations using the same variables. consistent system = a system of equations that has at least one solution.
independent system = a system of equations that has exactly one solution. dependent system = a system of equations that has an infinite number of solutions. inconsistent system = a system of equations that has no solution.
If a system of equations has only linear equations, those lines can be related in one of three ways: intersecting (one solution) parallel (no solution) coinciding (infinite solutions)
consistent, independent consistent, dependent inconsistent
Solving by Graphing Systems of equations can be solved by graphing each equation on the same plane. The point where the equations intersect is the solution to the system of equations.
Solving Systems of Equations Algebraically (3.2) Systems of equations can be solved using algebra techniques. There are two algebraic methods that are commonly used: Substitution Method Elimination Method
Substitution Method Þ Write one of the equations in terms of one of the variables. Þ Substitute the expression that the variable equals into the other equation. Þ Solve for the remaining variable. Þ Use that answer to solve for the other variable.
Elimination Method Þ Multiply at least one of the equations by a number that will make the coefficients of one of the variables opposites. (opposites = the same number but with different signs) Þ Add the equations together. Þ Solve for the remaining variable. Þ Use that answer to solve for the other variable.
When solving systems of equations algebraically, it is not as easy to see when there is no solution or when there are infinite solutions. When all the variables cancel out, the sentence that is left will be either true or false. If the sentence is true-->inf. sol. If the sentence is false--> no sol.
Solving Systems of Equations in Three Variables (3.7) Some situations have more than two variables. If a system of equations has three variables, there must be three equations to solve it. The solutions to a system of equations in 3 variables are called ordered triples.
Use the algebraic methods to solve systems in 3 variables. Þ Eliminate one of the variables two different ways using each equation at least once. (That will leave a system of equations in two variables.) Þ Then solve the 2-variable system Þ Solve for the entire ordered triple.
Solving Problems Using Systems Systems of equations can be used to solve word problems. u Identify the variables. u Change the words into equations. u Solve the system of equations. u Answer the question.
Some common types of word problems that use systems to solve are: u Coin/Value problems u Interest/Investment problems u Water/Current problems u Airspeed/Windspeed problems
Linear Inequalities (2.7) To graph a linear inequality: First, treat it like an equation y Solve for y. (Slope-intercept form) Graph the boundary line.* Shade above or below the line.** Use a test point to check.
*Graph the boundary line: The boundary line will be either a solid line or a dotted line. If the inequality uses or , the boundary line is dotted. If the inequality uses or , the boundary line is solid.
**Shade above or below the line: If the inequality uses or , shade above the line. If the inequality uses or , shade below the line.
> 2x + 3y > 6 > 3y > -2x + 6 > y > -2x/3 + 2
> 2x - y > 5 > -y > -2x + 5 < y < 2x - 5
y < 2 x < 2
Graphing Systems of Inequalities (3.4) A system of linear inequalities consists of more than one inequality to be graphed on the same coordinate plane. The shaded region will represent ordered pairs that make all the inequalities true.
To graph a system of inequalities: u Graph the boundary line for each inequality. u Determine where each would be shaded. u Shade ONLY where the shaded areas overlap. u Use a test point to check.
> y > x - 6 > y > -2x + 3
x - y < 3 x + y > 1 y > x - 3 y > -x + 1
y < 2x y < 3 - x/2 y > x - 3
| y | > 2 x < 3 y > 2 or y < -2 x < 3
Linear Programming (3.5) linear programming = a process used to find the maximum or minimum value of a linear function that is subject to given conditions on the variables.
In a linear programming problem, the conditions on the variables, or constraints, will be a system of inequalities. The shaded region for the system is called the feasible region.
If a linear programming problem has a maximum or minimum value, it will be at one of the vertices of the feasible region.
f(x,y) = 2x + 3y x > 1 y > 2 x + 2y < 9 (1,4) (5,2) (1,2) f(x,y) = 2x + 3y f(1,4) = 2(1) + 3(4) = 14 f(5,2) = 2(5) + 3(2) = 16 f(1,2) = 2(1) + 3(2) = 8 max = 16 min = 8
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