0.3.2 Rearranging equations

Do the same thing to both sides

An equation says two things are equal. It stays true if you do the same thing to both sides: add the same number, subtract it, multiply by it, or divide by it (as long as it is not zero).

To solve for $x$ , undo what was done to $x$ , in reverse order. In $3x + 5 = 20$ , the $x$ was multiplied by 3 and then had 5 added. So take away 5 first, then divide by 3.

1
$3x + 5 = 20$
2
$3x = 15$
3
$x = 5$

Rule: Balancing an equation

\begin{gathered} a = b \quad\Longrightarrow\quad a + c = b + c \\ a = b \quad\Longrightarrow\quad ac = bc \end{gathered}

Whatever you do to one side, do to the other. Undo the operations on the unknown in reverse order. Never divide by something that could be zero.

Try it Always check an answer by putting it back into the original equation. Both sides should match.

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The unknown on both sides

When $x$ appears on both sides, first gather every $x$ term on one side and every plain number on the other. In $5x - 3 = 2x + 9$ , subtract $2x$ from both sides to get $3x - 3 = 9$ . Then add 3 to get $3x = 12$ , so $x = 4$ .

Clear fractions first

Fractions make every later step harder. Multiply both sides by a number that every denominator divides into, and they disappear. In $\frac{x}{3} + \frac{x}{4} = 7$ , both 3 and 4 go into 12:

\begin{aligned} 12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{4} &= 12 \cdot 7 \\ 4x + 3x &= 84 \end{aligned}

So $7x = 84$ and $x = 12$ . Multiply every term by 12, including the 7 on the right.

Try it Check both answers by substitution. The second line uses Fraction from lesson 0.3.1, so the check is exact rather than a decimal close to 7.

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Making a letter the subject

A formula with several letters can be rearranged to give any one of them on its own. That letter is then the subject. The method is the same as solving for $x$ ; the other letters are treated like numbers.

Take $y = mx + c$ , the equation of a line (lesson 1.2.1). To make $x$ the subject, subtract $c$ , then divide by $m$ :

y - c = mx \quad\Longrightarrow\quad x = \frac{y - c}{m}

Sometimes the letter you want appears in two terms. Gather those terms on one side, then factor the letter out. Here is $ax + b = cx + d$ solved for $x$ :

1
$ax - cx = d - b$
2
$x(a - c) = d - b$
3
$x = \frac{d - b}{a - c}$

Rule: Making a letter the subject

\begin{gathered} \text{clear fractions} \;\to\; \text{gather} \\ \to\; \text{factor} \;\to\; \text{divide} \end{gathered}

Clear fractions. Gather the terms containing the letter on one side. Factor the letter out. Divide by what multiplies it, and note that this must not be zero.

Case study 0.3.5 uses exactly this to turn one interest formula into four.

Let SymPy check your algebra

\frac{y - c}{m}

import sympy as sp x, y, m, c, a, b, d = sp.symbols("x y m c a b d") print(sp.solve(sp.Eq(3*x + 5, 20), x)) print(sp.solve(sp.Eq(y, m*x + c), x)) print(sp.solve(sp.Eq(a*x + b, c*x + d), x))

⌘/Ctrl + Enter runs the cell

Common mistakes

\frac{2x + 6}{2}

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Solving and rearranging

0 of 8 correct

1.
Solve $4x - 7 = 13$
2.
Solve $2(x + 3) = 14$
3.
Solve $7x + 2 = 3x + 18$
4.
Solve $\dfrac{x}{2} + \dfrac{x}{3} = 10$
5.
Make $x$ the subject of $y = 3x - 6$
6.
Make $b$ the subject of $s = \dfrac{a + b}{2}$
7.
Make $x$ the subject of $ax + b = c$
8.
Make $p$ the subject of $q = p + pk$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Summary card Do the same thing to both sides. Undo the operations on the unknown in reverse order. Gather the unknown on one side; clear fractions by multiplying every term by a common denominator. To make a letter the subject: clear fractions, gather, factor it out, divide. The divisor must not be zero. Check every answer by substitution, or with sp.solve(sp.Eq(left, right), letter).

Do the same thing to both sides

Rule: Balancing an equation

The unknown on both sides

Clear fractions first

Making a letter the subject

Rule: Making a letter the subject

Let SymPy check your algebra

Common mistakes

Summary card

Free further reading