How to solve multivariable equations for one variable in terms of the others
How to solve multivariable equations
In this lesson we’ll look at how to solve a multivariable equation for a certain variable in terms of the others.
When you solve an equation for a variable you’re moving the other terms and coefficients around by using inverse operations to isolate the variable you’re solving for.
For example, if you have ???y=z+w+x??? and you want to solve the equation for ???w??? in terms of ???x,y,z??? then you need to move the terms so that the ???w??? is by itself on one side of the equation.
???y=z+w+x???
Move the ???z??? and the ???x??? by subtracting them from both sides.
???y-z-x=z-z+w+x-x???
???y-z-x=w???
You can rewrite ???w??? on the left but you don’t have to.
???w=y-z-x???
This is called isolating the variable ???w???.
Solving the same equation for three different variables
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Solving an equation for one variable in terms of all the others
Example
Solve for ???t??? in terms of ???w, x, y??? and ???z???.
???wx=-tvz???
We want to get ???t??? by itself on one side of the equation.
We can do this by dividing both sides by ???-1???, then dividing both sides by ???vz???.
???wx=-tvz???
???\frac{wx}{-1}=\frac{-tvz}{-1}???
???-wx=tvz???
???\frac{-wx}{vz}=\frac{tvz}{vz}???
???\frac{-wx}{vz}=t???
If you like you can move the ???t??? to the left hand side, and write the negative sign in front of the fraction.
???t=-\frac{wx}{vz}???
Let’s do another one with numerical coefficients.
Example
Isolate the variable ???s???.
???2s-3t+2u=7???
We need to get ???s??? by itself on one side of the equals sign. Let’s move the ???2u??? first.
???2s-3t+2u=7???
???2s-3t+2u-2u=7-2u???
???2s-3t=7-2u???
Move the ???-3t???.
???2s-3t+3t=7-2u+3t???
???2s=7-2u+3t???
Divide both sides by ???2???.
???\frac{2s}{2}=\frac{7-2u+3t}{2}???
???s=\frac{7-2u+3t}{2}???
Let’s do one more with numerical coefficients and variables on both sides.
Example
Solve for ???b??? in terms of ???a??? and ???c???.
???3a-4c-b=5a-3c???
We need to get ???b??? by itself on one side of the equals sign. Let’s move the ???3a??? first.
???3a-4c-b=5a-3c???
???3a-3a-4c-b=5a-3a-3c???
???-4c-b=2a-3c???
Move the ???-4c???.
???-4c+4c-b=2a-3c+4c???
???-b=2a+c???
Multiply by ???-1??? to isolate ???b???.
???-1(-b)=-1(2a+c)???
???b=-2a-c???