Solving for a Variable: Practice

Both terms on the left contain $x$, so factor it out, then divide by what is left.

$x(a + b) = c, \qquad x = \frac{c}{a + b}$

The factor is the sum $a + b$; dividing by $a$ alone would leave $x$ hiding in the second term.

Subtract $bx$ from both sides to gather the $x$ terms, then factor and divide.

$ax - bx = c, \qquad x(a - b) = c, \qquad x = \frac{c}{a - b}$

Both terms on the left contain $b$, so factor it out, then divide.

$b(a + c) = d, \qquad b = \frac{d}{a + c}$

Subtract $b$ from both sides, then divide the whole numerator by $x$.

$mx = y - b, \qquad m = \frac{y - b}{x}$

Gather the $x$ terms on one side and the constants on the other, then factor.

$ax - bx = a - b, \qquad x(a - b) = a - b, \qquad x = 1$

Dividing both sides by $a - b$ leaves $x = 1$, which is valid whenever $a \neq b$.

Subtract $c$ from both sides, then factor $x$ from the two $x$ terms and divide.

$ax - bx = d - c, \qquad x(a - b) = d - c, \qquad x = \frac{d - c}{a - b}$

Distribute on the right, then gather the $x$ terms on the left and factor.

$mx + n = px + pq, \qquad x(m - p) = pq - n, \qquad x = \frac{pq - n}{m - p}$

Combine the fractions on the left over the common denominator $ab$, then take the reciprocal of both sides.

$\frac{1}{f} = \frac{a + b}{ab}, \qquad f = \frac{ab}{a + b}$

Multiply both sides by $b$ to clear the denominator, then divide by $a$.

$by = ax, \qquad x = \frac{by}{a}$

Divide both sides by $n$ to free the parenthesis, then subtract $a$.

$a + l = \frac{2s}{n}, \qquad l = \frac{2s}{n} - a$

Distribute on the right, then gather the $x$ terms on the left and factor.

$ax + b = cx + cd, \qquad x(a - c) = cd - b, \qquad x = \frac{cd - b}{a - c}$

Add $ux$ to both sides and subtract $s$ to gather the $x$ terms, then factor and divide.

$rx + ux = t - s, \qquad x(r + u) = t - s, \qquad x = \frac{t - s}{r + u}$