Systems+of+Equations+&+Inequalities


 * 1.** **__ System of Linear Equations __**- a system made up of two or more linear equations.
 * [[image:http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/293px-Intersecting_Lines.svg.png width="293" height="264" caption="File:Intersecting Lines.svg" link="http://upload.wikimedia.org/wikipedia/commons/c/c0/Intersecting_Lines.svg"]] ||
 * File:Intersecting Lines.svg ||

**2.** **__ Solution of the System of Linear Equations __**- any ordered pair in a system that makes all the equations true.

**3.** **__ No Solution __**- when a system of linear equations have a graph with parallel lines.


 * 4.** **__ Infinitely Many Solutions __**- when a system of linear equations have a graph with the same line




 * 5//.//** __** Substitution Method **__- a method of solving a system of equations wherein one of the equations is solved for one variable in terms of the other variables
 * Step 1:** 5//x// + 3//y// = 26 [Equation 1.]
 * Step 2:** //x// - 3//y// = 4 [Equation 2.]
 * Step 3:** //x// = 3//y// + 4 [Rearrange equation 2.]
 * Step 4:** 5(3//y// + 4) + 3//y// = 26 [Substitute the values.]
 * Step 5:** 18//y// + 20 = 26 [Group the like terms.]
 * Step 6:** 18//y// = 6 [Subtracting 20 from the two sides of the equation.]
 * Step 7:** [[image:http://www.icoachmath.com/Sitemap/images/Substitution%20Method5.gif width="37" height="33" caption="external image Substitution%20Method5.gif"]] [Divide throughout by 18.]
 * Step 8:** [[image:http://www.icoachmath.com/Sitemap/images/Substitution%20Method6.gif width="125" height="33" caption="external image Substitution%20Method6.gif"]] [Substitute the values.]
 * Step 9:** //x// = 5 [Simplify.]
 * Step 10:** The solution for the linear system is [[image:http://www.icoachmath.com/Sitemap/images/Substitution%20Method1.gif width="36" height="33" caption="external image Substitution%20Method1.gif"]].


 * 6.** **__ Elimination Method __**- the process of eliminating one of the variables in a system of equations using addition or subtraction in conjunction with multiplication or division and solving the system of equations.
 * Step 1:** [[image:http://www.icoachmath.com/Sitemap/images/Elimination%20Method1.gif width="107" height="79" caption="external image Elimination%20Method1.gif"]] [Add the first two equations to eliminate //y//.]
 * Step 2:** [[image:http://www.icoachmath.com/Sitemap/images/Elimination%20Method2.gif width="92" height="79" caption="external image Elimination%20Method2.gif"]] [Add the second and third equations to eliminate //y//.]
 * Step 3:** We now have a system of two equations in two variables:
 * Step 4:** [[image:http://www.icoachmath.com/Sitemap/images/Elimination%20Method4.gif width="117" height="97" caption="external image Elimination%20Method4.gif"]] [Multiply the second equation with - 2 and add, and then solve for //x//.]
 * Step 5:** So, 3//x// + 4//z// = 15 gives //z// = 3 [Substitute the values.]
 * Step 6:** //x// + //y// + //z// = 6 gives //y// = 2 [Substitute the values.]
 * Step 7:** So, the solution is (1, 2, 3)**.**

y < x - 4
 * 7.** **__ Linear Inequality __**- involves a linear expression in two variables by using any of the relational symbols such as <, >, ≤ or ≥. A linear inequality divides a plane into two parts, if the boundary line is solid, then the linear inequality must be either ≥ or ≤, but if the boundary line is dotted, then the linear inequality must be either > or <.




 * 8.**  **__Solutions of an inequality__** - a number which when substituted for the variable makes the inequality a true statement.


 * 9. __ System of Linear Inequalities __**- two or more linear inequalities on the same plane


 * 10.** **__ Solution of a System of Linear Inequalities __**- when all the inequalities work, the region where all three individual solution regions overlap

^ solution set ^
 * 11.** **__ Solution Set __**- the set of values that satisfy a given set of equations or inequalities