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Algebra - Simultaneous Equations
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Algebra - Simultaneous Equations








































Solve Simultaneous Equations

Simultaneous equations are a set of two or more equations with multiple variables. (This is often referred to as a system of equations.)

Solve the following two equations for x and y.

Use the substitution, elimination (addition/subtraction), or graphical method

y = 1/3x - 2, y = -1/3x +1

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Feb 11, 2013
Solve Simultaneous Equations
by: Staff


Answer

Part I

Use the substitution, elimination (addition/subtraction), or graphical method

y = 1/3x - 2, y = -1/3x +1


Solving for x and y using substitution

The goal of the substitution method is to eliminate one of the variables using substitution.

y = ⅓ x - 2

y = -⅓ x +1

Since the 1st equation shows that y = ⅓ x - 2, substitute ⅓ x - 2 for y in the 2nd equation.

⅓ x - 2 = -⅓ x +1

The substitution transforms the 2nd equation into an equation which contains only one variable (the variable x).

Now solve for x.

⅓ x - 2 = -⅓ x +1

Add ⅓ x to each side of the equation.

⅓ x - 2 + ⅓ x = -⅓ x + 1 + ⅓ x

combine like terms

⅓ x + ⅓ x - 2 = -⅓ x + ⅓ x + 1

(⅓ x + ⅓ x) - 2 = (-⅓ x + ⅓ x) + 1

⅔ x - 2 = 0 + 1

⅔ x - 2 = 1

Multiply each side of the equation by 3

3 * (⅔ x - 2) = 3 * 1

3 * (⅔ x) + 3 * ( - 2) = 3 * 1

3 * (⅔ x) + 3 * ( - 2) = 3 * 1

2 * (3 / 3) * x + 3 * ( - 2) = 3 * 1

2 * (1) * x + 3 * ( - 2) = 3 * 1

2 x + 3 * ( - 2) = 3 * 1

2 x - 6 = 3 * 1

2 x - 6 = 3

Add 6 to each side of the equation

2 x - 6 + 6 = 3 + 6

2 x + 0 = 3 + 6

2 x = 3 + 6

2 x = 9

Divide each side of the equation by 2

2 x / 2 = 9 / 2

x * (2 / 2) = 9 / 2

x * (1) = 9 / 2

x = 9 / 2

x = 4.5

Now that you know that the value of x = 4.5, substitute 4.5 for x in either of the original two equations. It does not matter which equation you choose.

y = ⅓ x - 2

y = ⅓ * 4.5 - 2

y = 1.5 - 2

y = -0.5


the final answer is:

x = 4.5
y = - 0.5


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Feb 11, 2013
Solve Simultaneous Equations
by: Staff


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Part II


Solving for x and y using the elimination (addition/subtraction) method

The reason for using the elimination method is that one of the variables can be eliminated by addition or subtraction.

y = ⅓ x - 2, y = -⅓ x +1

Line up the two equations up vertically (one under the other).

y = ⅓ x - 2

y = -⅓ x +1

The 1st equation contains the positive term ⅓ x. The second equation contains the term -⅓ x.

When these two equations are added, these terms will cancel one another. This completely eliminates the x variable and allows you with only one equation with one variable (the variable y) .

y = ⅓ x - 2

+(y = -⅓ x +1)
-----------------------
(y + y) = (⅓ x - ⅓ x) + (- 2 + 1)

2y = 0 - 1

2y = - 1

The next step is to divide each side of the equation by 2

2y / 2 = - 1 / 2

y * (2 / 2) = - ½

y * (1) = - ½

y = - ½

Now that you know that the value of y = - ½, substitute - ½ for y in either of the original two equations. It does not matter which equation you choose.

y = ⅓ x - 2

- ½ = ⅓ x - 2

You now have an equation containing only the variable x.

Solve for x

- ½ = ⅓ x - 2

multiply each side of the equation by 6

6 * (- ½) = 6 * (⅓ x - 2)

- (6/2) = (6/3) x - (6 * 2)

- 3 = 2 x - 12

2 x - 12 = - 3

Add 12 to each side of the equation

2 x - 12 + 12 = - 3 + 12

2 x + 0 = - 3 + 12

2 x = - 3 + 12

2 x = 9

Divide each side of the equation by 2

2 x / 2 = 9 / 2

x * (2 / 2) = 9 / 2

x * (1) = 9 / 2

x = 9 / 2

x = 4.5

the final answer is:

x = 4.5
y = - 0.5



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Feb 11, 2013
Solve Simultaneous Equations
by: Staff


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Part III



Solving for x and y using the graphical method

Plot both of the equations on the same set of x-y coordinates.:

y = ⅓ x - 2

y = -⅓ x +1

The solution values of x and y is that point where both graphs intersect.
Solve Simultaneous Equations










Thanks for writing.

Staff
www.solving-math-problems.com



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