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Solve both equations for y.

 * 1) Graph each equation using y = mx + b.
 * 2) Determine the solution by the graph.
 * If the two lines intersect, the solution is the point where the two lines intersect.
 * If the two lines are parallel, the solution is NO SOLUTION.
 * If the two lines overlap, the solution is infinitely many solutions.
 * ==Examples: 2x - 3y = 1 and x + y = 3 3x - 2y = 6 and==

==Word Problem: Your family is planning a 7 day trip to Arizona. You estimate that it will cost $300 per day in Phoenix and $675 per day in the Grand Canyon. Your total budget for the 7 days is $2850. How many days should you spend in each location?==

time spent in Phoenix + time spent in Grand Canyon = total vacation time (x: Phoenix, y: Grand Canyon) 300x x 675y = 2850
__Substitution__

4. Substitute the values for both variables into both equations to show they are correct.
__Example:__ //y// – 3//x// = 5 and //y// + //x// = 3 To isolate the //y// variable we subtract //x// from both sides. ==//y// + //x// = 3 //y// + //x// – //x// = 3 – //x y// = 3 – //x// In equation //y// – 3//x// = 5, replace the variable //y// with the value for //y// obtained in step 1, from y+x=3. //y// – 3//x// = 5 (3 – //x//) – 3//x// = 5==

==Solve for //x// by first combining like terms, then //isolating// the terms with the //x// variable. 3 – //x// – 3//x// = 5 3 – 3 – 4//x//= 5 – 3 – 4//x// = 2 Divide both sides by the coefficient of //x// to //isolate// //x//. //x// = 2/–4 = –½==

Now we substitute the value of //x// = –½ and //y// = 3½ into both of our original equations.
y – 3//x// = 5 3½ – 3(–½) = 5 3½ + 1½ = 5 5 = 5 //y// + //x// = 3 3½ + (–½) = 3 3 = 3

__Word Problem:__ How many grams of pure gold and how many grams of an alloy that is 55% gold should be melted together to produce 72 g of an alloy that is 65% gold? x = =

grams of pure gold y grams of the alloy x + y = 72 x + .55y = .65(72) = 46.8 x =72 - y=

(72 - y) + .55y 46.8 72 - .45y = 46.8 -.45y = -25.2 y =56=

y 56 x == =72 - 56=

16

Approximately 16 grams alloy and 56 grams of pure gold need to be used in order to have 72 g of .55 alloy.


 * Elimination:**

Step 1: Line up the equations so that the variables are lined up vertically. Step 2: Choose the easiest variable to eliminate and multiply both equations by different numbers so that the coefficients of that variable are the same. Step 3: Subtract the two equations. Step 4: Solve the one variable system. Step 5: Put that value back into either equation to find the other equation.

Step 6: Reread the question and plug your answers back in to check. Example Solve 2x = 3y + 3 4x - 5y = 7

**Solution** 4x - 5y = 7 > 4x - 6y = 6 4x - 5y = 7 > > > > >    4x = 12 x = 3 > > We see that 2(3) = 3(1) + 3   4(3) - 5(1) = 7
 * 1)  2x - 3y = 3
 * 1) Multiply the first equation by 2.
 * 1)  -y = -1 After subtracting the equations.
 * 1)  y = 1
 * 1)  4x - 5(1) = 7 Substituting 1 for y in the second equation.
 * 1) The answer is (3,1)

Solving Systems of Inequalities

**Example** Graph the system of inequalities: 3x + y __>__ 12 3x + 2y __<__ 15 y __>__ 2

**Solution** We draw T-tables to graph the two lines. Note that the last two lines is horizontal. 3x + y = 12 || ** x ** || ** y ** || 3x + 2y = 15 We solve the two by two system to find the coordinates of the intersection. y = 12 - 3x 3x + 2(12 - 3x) = 15 3x + 24 - 6x = 15 -3x = -9 x = 3 Plugging back in y =12 - 3(3)= 3 Hence the point (3,3) is the point of intersection. The graph is shown below.
 * 0 || 12 ||
 * 4 || 0 ||
 * ** x ** || ** y ** ||
 * 0 || 7.5 ||
 * 5 || 0 ||

Link: http://www.ltcconline.net/greenl/Courses/152b/QuadraticsLineIneq/systems.htm


 * Key Words:**


 * Linear:** of, relating to, resembling, or having a graph that is a line and especially a straight line


 * Graphing:** to represent by a graph


 * Subsittution:** replacement of one mathematical entity by another of equal value


 * Inequalites:** a formal statement of inequality between two quantities usually separated by a sign of inequality (as <, >, or ≠ signifying respectively //is less than, is greater than,// or //is not equal to//)


 * Systems:** a regularly interacting or interdependent group of items forming a unified whole number system