Difference between revisions of "Substitution"

m (Added latex and fixed typo.)
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<math>x+y=-1</math>          Subtract <math>x</math> from both sides.   
 
<math>x+y=-1</math>          Subtract <math>x</math> from both sides.   
 +
 
<math>y=-x-1</math>          <math>y</math> is now isolated.
 
<math>y=-x-1</math>          <math>y</math> is now isolated.
  
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<math>3x-(-x-1)=5 </math>    Distribute the negative sign.   
 
<math>3x-(-x-1)=5 </math>    Distribute the negative sign.   
 +
 
<math>3x+x+1=5</math>        Combine like terms.
 
<math>3x+x+1=5</math>        Combine like terms.
 +
 
<math>4x+1=5</math>          Subtract 1 from both sides.
 
<math>4x+1=5</math>          Subtract 1 from both sides.
 +
 
<math>4x=4</math>            Divide both sides by four.
 
<math>4x=4</math>            Divide both sides by four.
 +
 
<math>x=1</math>             
 
<math>x=1</math>             
  
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1+y=-1          Subtract 1 from both sides.
 
1+y=-1          Subtract 1 from both sides.
 +
 
y=-2
 
y=-2
  
(x,y)=(1,-2)
+
<math>(x,y)=(1,-2)</math>
  
 
You can check this answer by plugging x and y into the original equations.   
 
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.
 
This same method is used for simultaneous equations with more than two equations.

Revision as of 21:57, 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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