Rearranging Equations

In this section we’re going to be looking at rearranging equations. We shall explore how to arrange linear equa...

In this section we’re going to be looking at rearranging equations. We shall explore how to arrange linear equations, brackets, fractions and quadratics. This entry explores the basic ideas of rearranging equations. You can find examples of harder rearranging equations here

What is the subject?

Consider the following SUVAT formula. The following formula is used to find the velocity of an object over time. In the formula V (velocity) is known as the subject of the formula. That is; VELOCITY is the subject we’re trying to solve or find.

In the formula v stands for final velocity, u stands for initial velocity, a stands for acceleration, and t stands for time.

The formula may be used used to find the velocity of an object given that you know the initial velocity of the object, its acceleration, and the time (how long it has moved for). Suppose we wanted to find time (t) in the formula, we would have to know what the value of v, u and a is.

We rearrange an equation to change the subject by making another variable the subject to be found. To find time we would have to rearrange the equation to make t the subject so that time can be found. There a few basic rules that you have to remember.

v = u + at

To make t the subject we have to make sure that t is the only letter left standing on one side of the equation.

What we do to one side we must also do to the other side
Step 1: Subtract u from both sides

We have to move u to the other side so that the term with a is left on its own. To do this we have to subtract u from both sides of the equation.

v - u = u - u + at

This operation should provide the following equation.

v – u = at
Step 2: Divide both sides by a

We must divide both sides by a to make sure that t is left on its own.

v – u/a = at/a

This operation should provide the following equation.

v – u/a = t

Lastly we rewrite the equation in a more easy readable form.

t = v – u/a

Rearranging equations with brackets

In this section we shall explore how to rearrange equations which involve brackets.

Example Make y the subject in the equation x=k(y+z)
Answer:
x/k = k(y + z)/k
y = x/k – z
Explanation

There are a few basic steps that we must follow in rearranging this equation. In this example we must rearrange the equation;

x=k(y+z)
Step 1: Divide both sides by k

There is a multiplication on one side of the equation. Since y is on the other side we have to find a way to leave it on its own. To get rid of k on the other side we must divide both sides by k.

x/k = k(y + z)/k

This operation provdes the following results;

x/k = y + z
Step 2: Subtract z from both sides

We have y one the other side but with another term or letter which we have to get rid of. To get rid of this we must subtract z from both sides.

x/k - z = y + z - z

This operation should provide the following results.

x/k – z = y

We rewrite it in an easy to approach form;

y = x/k – z

We have managed to make y the subject

Rearranging equations with fractions

This section explores how to rearrange equations with fractions.

Example Make x the subject of the following equation;
a + 4/x = 5b
Answer:
a + 4/x = 5b/1
a + 4 = 5bx
a + 4/5b = 5bx/5b
x = a + 4/5b
Explanation

Note that;

a + 4/x = 5b/1
Always get rid of the fractions
Step 1: Multiply both sides by x

It is a good idea to get rid of the fractions first. We do this by multiplying both sides by x..

a + 4/x × x = 5b × x

This shoulf provide the following results;

a + 4 = 5bx
Step 3: Divide both sides by 5b

Since we’re trying to make x the subject we have to get rid of 5b, we do this by dividing both sides by 5b.

a + 4/5b = 5bx/5b

This operation should provide the following result.

a + 4/5b = x

This can be rewritten as;

x = a + 4/5b

Rearranging equations with quadratics

The following shows an equation which involves quadratics;

Example Make x the subject in the following equation.
ax² + f/e = b
Answer:
ax² + f = be
x² = be – f/a
x = √be – f/a
Explanation

This example contains a sqaure which might make the rearranging seem complicated. But all we have to do is undo the square.

Always get rid of the fractions first.
Step 1: Multiply both sides by e

Getting rid of the fraction should simplify the equation. We can do this by multiplying both sides by e

ax² + f/e × e= b × e

This operation should provide the following result.

ax² + f = be
Step 2:Subtract f from both sides

We must leave x2 on its own since we’re trying to make x the subject. To do this we must get rid of f on the left hand of the equation..

ax² = be – f
Step 3: Divide both sides by a

The letter x is not yet alone. We still have to get rid of a from the left hand side. To get rid of a we must divide both sides by a.

ax²/a = be – f/a

This operation should provide the following result.

x² = be – f/a
Step 4: Square root both sides

To make x2 x we must square root it, this means we must square root both sides if both sides of the equal are to remain equal.

√x² = √be – f/a

This operation should provide the following answer.

x = √be – f/a
Example: Rearrange the following expression by making x the subject.
a/(x – b)2 = c
Answer:
    Explanation:
    Step 1: Multiply both sides by (x – b)²

    First we multiple each side with (x – b)2 to get rid of the fractions.

    a/(x – b)² × (x – b)²= c × (x – b)²

    This operation should provide the following results.

    a = c(x – b)2
    Don’t expand! When working with powers you never expand. You simply undo the operation on both sides of the equation.
    Step 2: Divide both sides by c

    Divide both sides with c so that you leave the term with x alone.

    a/c = c(x – b)²/c
    a/c = (x – b)²

    This operation should provide the following results.

    a/c = (x – b)²
    Step 3: Square root both sides

    To get rid of the square in (x – b)2 we have to square root it. What we do to one side we must also do to the other side.

    a/c = √((x – b)²)

    This operation should provide the following result.

    a/c = x – b
    Step 4: Add b to both sides.

    The letter should be the only letter left standing on one side of the equation. There is still the letter b on the right hand side of the equation. To get rid of it we must add b to both sides of the equation.

    a/c + b = x – b + b

    This operation should provide the following result.

    x = b + √a/c
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