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.

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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.

$t=-\frac{wx}{vz}$

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Let’s do another one with numerical coefficients.

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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.

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 $2$.

$\frac{2s}{2}=\frac{7-2u+3t}{2}$

$s=\frac{7-2u+3t}{2}$

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Let’s do one more with numerical coefficients and variables on both sides.

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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 $b$.

$-1(-b)=-1(2a+c)$

$b=-2a-c$

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