Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Substitution"

m (Added latex and fixed typo.)
(Cleanup and added clever substitution section.)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
 +
'''Substitution''' is when one replaces all instances of a variable (or expression) with another equivalent variable (or expression).
 +
 +
==Uses==
 +
 +
===System of Equations===
 +
 +
{{main|System of equations}}
 +
 
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:
 
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
 
1. Isolation of a variable
 +
 
2. Substitution of variable into another equation to reduce the number of variables by one
 
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.   
 
3. Repeat until there is a single equation in one variable, which can be solved by means of other methods.   
  
−
Example:
+
For example, consider the below system.
 +
<cmath>\left\{\begin{array}{l}x+y=-1\\3x-y=5\end{array}\right.</cmath>
 +
An example of solving the system by substitution is when we start by isolating <math>y</math> in the top equation to get <math>y = -x - 1</math>.  Then, we can replace all instances of <math>y</math> with <math>(-x-1)</math> in the second equation.  Doing so results in an equation with one variable, and solving it results in
 +
<cmath>\begin{align*}
 +
3x-(-x-1) &= 5 \\
 +
3x+x+1 &= 5 \\
 +
4x+1 &= 5 \\
 +
4x &= 4 \\
 +
x &= 1.
 +
\end{align*}</cmath>
 +
After solving for <math>x</math>, we can "plug in" the value <math>1</math> for <math>x</math> to get <math>y = -(1)-1 = -2</math>, so the solution to the system is <math>(1,-2)</math>.  As usual, we can check by substituting <math>1</math> for <math>x</math> and <math>-2</math> for <math>y</math>.
 +
 
 +
This same method is used for simultaneous equations with more than two equations.
  
−
Solve <math>x+y=-1, 3x-y=5</math> for <math>(x,y)</math>. 
+
===Observing Common Parts===
  
−
Start with <math>x+y=-1</math>.
+
Substitution can also be used when an expression has multiple common parts.
  
−
<math>x+y=-1</math>         Subtract <math>x</math> from both sides.   
+
For instance, consider the equation <math>4^x - 6 \cdot 2^x + 8 = 0</math>.  Note that with exponent properties, we can rewrite the equation as <math>(2^x)^2 - 6 \cdot 2^x + 8 = 0</math>.  Because there are multiple instances of <math>2^x</math> in the equation, we can let <math>y = 2^x</math> and substitute <math>2^x</math> for <math>y</math> to make the equation easier to solve.
−
<math>y=-x-1</math>         <math>y</math> is now isolated.
 
  
−
Substitute <math>(-x-1)</math> for the y in <math>3x-y=5.</math>
+
Doing so results in
 +
<cmath>\begin{align*}
 +
y^2 - 6y + 8 &= 0 \\
 +
(y-4)(y-2) &= 0 \\
 +
y &= 2, 4.
 +
\end{align*}</cmath>
 +
Finally, we can substitute <math>y</math> for <math>2^x</math> to get <math>2^x = 2</math> or <math>2^x = 4</math>, resulting in <math>x = 1</math> or <math>x = 2</math>. As usual, we can check by substituting <math>1</math> and <math>2</math> for <math>x</math> separately.
  
−
<math>3x-(-x-1)=5 </math>    Distribute the negative sign. 
+
==Problems==
−
<math>3x+x+1=5</math>        Combine like terms.
 
−
<math>4x+1=5</math>          Subtract 1 from both sides.
 
−
<math>4x=4</math>            Divide both sides by four.
 
−
<math>x=1</math>           
 
  
−
x is now solved for, so substitute x into one of the original equations. 
+
===Introductory===
  
−
1+y=-1          Subtract 1 from both sides.
+
* [[1961 AHSME Problems/Problem 5]]
−
y=-2
+
* [[1969 AHSME Problems/Problem 17]]
 +
* [[2007 iTest Problems/Problem 2]]
  
−
(x,y)=(1,-2)
+
===Intermediate===
  
−
You can check this answer by plugging x and y into the original equations. 
+
* [[1983 AIME Problems/Problem 3]]
  
−
This same method is used for simultaneous equations with more than two equations.
+
[[Category:Algebra]]

Latest revision as of 15:53, 10 April 2020

Substitution is when one replaces all instances of a variable (or expression) with another equivalent variable (or expression).

Uses

System of Equations

Main article: System of equations

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.

For example, consider the below system. \[\left\{\begin{array}{l}x+y=-1\\3x-y=5\end{array}\right.\] An example of solving the system by substitution is when we start by isolating $y$ in the top equation to get $y = -x - 1$. Then, we can replace all instances of $y$ with $(-x-1)$ in the second equation. Doing so results in an equation with one variable, and solving it results in \begin{align*} 3x-(-x-1) &= 5 \\ 3x+x+1 &= 5 \\ 4x+1 &= 5 \\ 4x &= 4 \\ x &= 1. \end{align*} After solving for $x$, we can "plug in" the value $1$ for $x$ to get $y = -(1)-1 = -2$, so the solution to the system is $(1,-2)$. As usual, we can check by substituting $1$ for $x$ and $-2$ for $y$.

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

Observing Common Parts

Substitution can also be used when an expression has multiple common parts.

For instance, consider the equation $4^x - 6 \cdot 2^x + 8 = 0$. Note that with exponent properties, we can rewrite the equation as $(2^x)^2 - 6 \cdot 2^x + 8 = 0$. Because there are multiple instances of $2^x$ in the equation, we can let $y = 2^x$ and substitute $2^x$ for $y$ to make the equation easier to solve.

Doing so results in \begin{align*} y^2 - 6y + 8 &= 0 \\ (y-4)(y-2) &= 0 \\ y &= 2, 4. \end{align*} Finally, we can substitute $y$ for $2^x$ to get $2^x = 2$ or $2^x = 4$, resulting in $x = 1$ or $x = 2$. As usual, we can check by substituting $1$ and $2$ for $x$ separately.

Problems

Introductory

Intermediate

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Substitution&oldid=120797"