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. The formula above is used to find velocity over time. Suppose we wanted to find time (t) in the formula, we would have to know what the value of v is. Let’s rearrange the formula below to make t the subject.
v = u + atFirst move u to the other side to leave the time (t) with the other letter a, we’re trying to make t the subject to do this we’ll have to make sure that t is on its own.
v – u = atNow to leave t on its own we must divide both sides by a. Now we have;
v – u/a = tLastly we rewrite the equation in a more easy readable form.
t = v – u/a
Rearranging equations with bracketsIn 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 In this example we must rearrange the equation;
x=k(y+z)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. That leaves;
x/k = y + zWe 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 – zThat leaves;
x/k – z = yWe rewrite it in an easy to approach form;
y = x/k – zWe have managed to make y the subject
Rearranging equations with fractionsIn the following example we’re going to look at rearranging equations with fractions.
Example Make x the subject of the following equation;
a + 4/x = 5b
We have to get rid of the fraction; we do this by multiplying both sides by x or we cross multiply. …we multiply 5b with x and a+4 with 1 which results in;
a + 4 = 5bxSince 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 leaves;
a + 4/5b = xRewrite it as;
x = a + 4/5bWe have managed to make x the subject.
Rearranging equations with quadraticsThe following shows an equation which involves quadratics;
Example Given that;
ax² + f/e = bMake x the subject
Answer In the above example we have a square. First we have to get rid of the fraction as we did before. We do this by dividing both sides by e or cross multipliying. We multiply e with b and ax2 + f with 1 to get the following result.
ax² + f = beNow we must leave x2 on its own since we’re trying to make x the subject. First we subtract f from both sides.
ax² = be – fNow we divide both sides by a as shown below; That leaves;
x² = be – f/aTo make x2 x we must square root it, this means we must square root both sides if they’re to remain equal.
√x² = √be – f/aThis results in;
x = √be – f/a