Solving Formulas for a Specified Variable

Introduction to Formulas

A formula is an equation that expresses a relationship between two or more variables. In algebra, you are often required to solve a formula for a specific variable, meaning you rearrange the equation so that the chosen variable is isolated on one side of the equation.

• Solving for a variable means making that variable the subject of the formula, with all other terms on the opposite side.

• This process is essential for manipulating equations in mathematics, science, and engineering.

Warm-Up: Solving Simple Equations

• Example 1: Solve $7 = 33x$ for $x$.

  • Divide both sides by 33:

  • $x = \frac{7}{33}$

• Example 2: Solve $12 = 23x - 5$ for $x$.

  • Add 5 to both sides: $12 + 5 = 23x$

  • $17 = 23x$

  • Divide both sides by 23: $x = \frac{17}{23}$

Solving Formulas: Step-by-Step Examples

• Example 3: Solve $V = lwh$ for $h$.

  • Divide both sides by $lw$:

  • $h = \frac{V}{lw}$

• Example 4: Solve $C = 59(F - 32)$ for $F$.

  • Divide both sides by 59: $\frac{C}{59} = F - 32$

  • Add 32 to both sides: $F = \frac{C}{59} + 32$

• Example 5: Solve $P = b + 1.5t$ for $t$.

  • Subtract $b$ from both sides: $P - b = 1.5t$

  • Divide by 1.5: $t = \frac{P - b}{1.5}$

• Example 6: Solve $2x + 3y = 7$ for $y$.

  • Subtract $2x$ from both sides: $3y = 7 - 2x$

  • Divide by 3: $y = \frac{7 - 2x}{3}$

• Example 7: Solve $y + mx_2 = mx_1$ for $m$.

  • Subtract $y$ from both sides: $mx_2 = mx_1 - y$

  • Bring all $m$ terms to one side: $mx_2 - mx_1 = -y$

  • Factor $m$: $m(x_2 - x_1) = -y$

  • Divide by $(x_2 - x_1)$: $m = \frac{-y}{x_2 - x_1}$

• Example 8: Solve $F = mv2r$ for $m$.

  • Assuming the formula is $F = m v^2 r$ (Additional info: clarified notation):

  • Divide both sides by $v^2 r$:

  • $m = \frac{F}{v^2 r}$

Practice Problems

• Example 9: Solve $C = 2\pi r$ for $r$.

  • Divide both sides by $2\pi$:

  • $r = \frac{C}{2\pi}$

• Example 10: Solve $S = 2WH + 2WL$ for $H$.

  • Subtract $2WL$ from both sides: $S - 2WL = 2WH$

  • Divide by $2W$: $H = \frac{S - 2WL}{2W}$

• Example 11: Solve $pr + pw = m$ for $p$.

  • Factor $p$ from both terms: $p(r + w) = m$

  • Divide by $(r + w)$: $p = \frac{m}{r + w}$

• Example 12: Solve $abc + b = 2a$ for $b$.

  • Subtract $2a$ from both sides: $abc + b - 2a = 0$

  • Factor $b$: $b(ac + 1) = 2a$

  • Divide by $(ac + 1)$: $b = \frac{2a}{ac + 1}$

• Example 13: Solve $xa + yb = 1$ for $x$.

  • Subtract $yb$ from both sides: $xa = 1 - yb$

  • Divide by $a$: $x = \frac{1 - yb}{a}$

Things to Remember and Mistakes to Avoid

• If the variable you are solving for appears in a fraction, use the multiplication principle to clear denominators.

• Isolate all terms containing the variable on one side of the equation.

• If the variable appears in more than one term, factor it out before solving.

• Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable.

Summary: Steps for Solving a Formula for a Given Variable

1. Clear fractions if the variable is in a denominator.

2. Isolate all terms containing the variable on one side.

3. Factor the variable if it appears in multiple terms.

4. Solve for the variable using appropriate algebraic operations.

Additional Practice Problems

• Example 1: Solve $I = Prt$ for $t$.

  • Divide both sides by $Pr$:

  • $t = \frac{I}{Pr}$

• Example 2: Solve $A = \frac{1}{2}bh$ for $b$.

  • Multiply both sides by 2: $2A = bh$

  • Divide by $h$: $b = \frac{2A}{h}$

• Example 3: Solve $p = 23a - 10$ for $a$.

  • Add 10 to both sides: $p + 10 = 23a$

  • Divide by 23: $a = \frac{p + 10}{23}$

• Example 4: Solve $x = ya + yz$ for $y$.

  • Factor $y$: $x = y(a + z)$

  • Divide by $(a + z)$: $y = \frac{x}{a + z}$

• Example 5: Solve $x + 7y = 1$ for $y$.

  • Subtract $x$ from both sides: $7y = 1 - x$

  • Divide by 7: $y = \frac{1 - x}{7}$

Practice these steps with a variety of formulas to build confidence and fluency in solving for any variable.