Difference between revisions of "Substitution"

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Start with <math>x+y=-1</math>.   
 
Start with <math>x+y=-1</math>.   
  
x+y=-1          Subtact x from both sides.   
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<math>x+y=-1</math>         Subtract <math>x</math> from both sides.   
y=-x-1          y is now isolated.
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<math>y=-x-1</math>         <math>y</math> is now isolated.
  
Substitute (-x-1) for the y in 3x-y=5.
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Substitute <math>(-x-1)</math> for the y in <math>3x-y=5.</math>
  
3x-(-x-1)=5     Distribute the negative sign.   
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<math>3x-(-x-1)=5 </math>    Distribute the negative sign.   
3x+x+1=5        Combine like terms.
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<math>3x+x+1=5</math>       Combine like terms.
4x+1=5          Subtract 1 from both sides.
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<math>4x+1=5</math>         Subtract 1 from both sides.
4x=4            Divide both sides by four.
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<math>4x=4</math>           Divide both sides by four.
x=1             
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<math>x=1</math>              
  
 
x is now solved for, so substitute x into one of the original equations.   
 
x is now solved for, so substitute x into one of the original equations.   

Revision as of 21:49, 22 April 2018

Substitution is a relatively universal method to solve simultaneous equations. It is generally introduced in a first year high school algebra class. A solution generally exists when the number of equations is exactly equal to the number of unknowns. The method of solving by substitution includes:

1. Isolation of a variable 2. Substitution of variable into another equation to reduce the number of variables by one 3. Repeat until there is a single equation in one variable, which can be solved by means of other methods.

Example:

Solve $x+y=-1, 3x-y=5$ for $(x,y)$.

Start with $x+y=-1$.

$x+y=-1$ Subtract $x$ from both sides. $y=-x-1$ $y$ is now isolated.

Substitute $(-x-1)$ for the y in $3x-y=5.$

$3x-(-x-1)=5$ Distribute the negative sign. $3x+x+1=5$ Combine like terms. $4x+1=5$ Subtract 1 from both sides. $4x=4$ Divide both sides by four. $x=1$

x is now solved for, so substitute x into one of the original equations.

1+y=-1 Subtract 1 from both sides. y=-2

(x,y)=(1,-2)

You can check this answer by plugging x and y into the original equations.

This same method is used for simultaneous equations with more than two equations.

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