Solving+Systems+of+Equations-+Substitution

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Algebra II
Substituion Of Equations By Using Substitution.

Defintions :

 * Equations : The act or process of equating or of being equated.
 * Substitution : The act or an instance of substituting & The state of being substituted.
 * Graphing : is a useful tool for solving systems of equations, but it can sometimes be time-consuming

Example 1 : Solve the following system, using substitution:

 * 5//x// + //y// || = || 13 ||  ||
 * 3//x// || = || 15 - 3//y// ||  ||


 * The easiest variable to isolate is //y// in the first equation, because it has no coefficient:
 * //y// = 13 - 5//x//
 * In the second equation, substitute for //y// its equivalent expression:
 * 3//x// = 15 - 3(13 - 5//x//)
 * Solve the equation:
 * 3//x// = 15 - 39 + 15//x//
 * 3//x// = 15//x// - 24
 * -12//x// = - 24
 * //x// = 2
 * Now substitute this //x// -value into the "isolation equation" to find //y// :
 * //y// = 13 - 5//x// = 13 - 5(2) = 13 - 10 = 3
 * Thus, the solution to the system is (2, 3) . It is useful to check this solution in both equations.
 * **Note:** Although we chose //y// in the first equation in the previous example, isolating any variable in any equation will yield the same solution.

Example 2 : Solve the following system, using substitution:

 * 2//x// + 4//y// || = || 36 ||  ||
 * 10//y// - 5//x// || = || 0 ||  ||


 * It is easier to work with the second equation, because there is no constant term:
 * 5//x// = 10//y//
 * //x// = 2//y//
 * In the first equation, substitute for //x// its equivalent expression:
 * 2(2//y//) + 4//y// = 36
 * Solve the equation:
 * 4//y// + 4//y// = 36
 * 8//y// = 36
 * //y// = 4.5
 * Plug this //y// -value into the isolation equation to find //x// :
 * //x// = 2//y// = 2(4.5) = 9
 * Thus, the solution to the system is (9, 4.5).

Example 3 : Solve the following system, using substitution:

 * 2//x// - 4//y// || = || 12 ||  ||
 * 3//x// || = || 21 + 6//y// ||  ||


 * It is easiest to isolate //x// in the second equation, since the //x// term already stands alone:
 * //x// = [[image:http://img.sparknotes.com/figures/A/ae9daa68f8a4f991e2068793c87afedc/latex_img6.gif]]
 * //x// = 7 + 2//y//
 * In the first equation, substitute for //x// its equivalent expression:
 * 2(7 + 2//y//) - 4//y// = 12
 * Solve the equation:
 * 14 + 4//y// - 4//y// = 12
 * 14 = 12
 * Since 14≠12, the system of equations has no solution. It is inconsistent (and independent). The two equations describe two parallel lines.

This Is an Example Of a Problem Without a Solution :


This Is an Example Of a Problem With a Solution :


media type="youtube" key="KNwwu5wjcaA" height="344" width="425"

QUIZ:

1) Use substitution to solve the system: 2) Set the two equations equal to calculate the solution to the system below: 3) Solve the system of linear equations by setting their equations equal: 4) Solve the system of linear equations by setting their equations equal: 5) Use the substitution method to solve the system:
 * Line 1: y = 3x + 1
 * Line 2: 4y = 12x + 3
 * Line 1: y= x + 1
 * Line 2: y= 2x
 * Line 1: y= x +5
 * Line 2: y= 2x +2
 * Line 1: y= x – 1
 * Line 2: y= 2x +2
 * Line 1: y = 5x – 1
 * Line 2: y= 3x + 12

Citation For Examples : Citation For Media Files : Substitution Powerpoint : Non Media Links : __Stapel, Elizabeth. "Solving One-Step Linear Equations." Purplemath. Available from__ __http://www.purplemath.com/modules/solvelin.htm. Accessed 04 December 2008__
 * __Solving Systems of Linear Equations by Substitution__. 14 Nov. 2008 http://www.sparknotes.com/math/algebra1/systemsofequations/section2.rhtml.
 * __Solving Systems of Linear Equations by Substitution__. 14 Nov. 2008 .
 * __Solving Systems of Linear Equations by Substitution__. 14 Nov. 2008 .
 * __Video of Substitution__. 18 Nov. 2008. .
 * [|[[http://cbhsmath.wikispaces.com/file/view/Systems_of_Linear_Equations.ppt|Systems_of_Linear_Equations.ppt]]]
 * http://www.purplemath.com/modules/solvelin.htm

Stapel, Elizabeth. "Systems of Linear Equations: Solving by Substitution." __Purplemath__. Available from __ http://www.purplemath.com/modules/systlin4.htm  __. Accessed 04 December 2008 Stapel, Elizabeth. "Other Number Properties: Identities, Inverses, Symmetry, etc." __Purplemath__. Available from __ http://www.purplemath.com/modules/numbprop2.htm  __. Accessed 04 December 2008
 * http://www.purplemath.com/modules/systlin4.htm
 * http://www.purplemath.com/modules/numbprop2.htm

QUIZ ANSWERS: 1) No solution 2) (1,2) 3) (3,8) 4) (-3,-4) 5) (2,9)