Solving for a Variable (Symbolic Isolation): Step-by-Step Guide
Sometimes you need to rearrange a formula to get a different letter by itself. For example, you might change d = rt into t = d/r. The steps are the same as solving a regular equation: pick the letter you want, treat all the other letters like numbers, and undo the operations one at a time, doing the same thing to both sides.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Pick the letter to solve for. Decide which letter you want by itself. Treat all the other letters as if they were numbers.\[\text{solve for } y \text{ in } f(x, y, z) = 0\]
- Clear fractions and parentheses. If the letter you want is in a denominator, multiply both sides by that denominator. Then multiply out any parentheses.\[\frac{P}{Q} = R \;\Rightarrow\; P = QR\]
- Gather the terms with your letter. Move every term that has your letter to one side. Move everything else to the other side.\[ay + by = c - dy \;\Rightarrow\; ay + by + dy = c\]
- Factor out your letter. If your letter shows up in more than one term, pull it out as a common factor, so it shows up only once.\[y(a + b + d) = c\]
- Divide and check. Divide both sides by whatever is multiplying your letter. To check, try some simple numbers in both the original and the new formula. They should agree.\[y = \frac{c}{a + b + d}\]
Worked example
Solve for x in terms of the other variables: a x + b = c x + d
Step 1: We want to solve for x. The letters a, b, c, and d act like numbers.
Step 2: There are no fractions or parentheses, so we can skip this step.
Step 3: Move the x terms to the left and the others to the right. Subtract cx and subtract b from both sides: ax - cx = d - b.
Step 4: Factor x out of the left side: x(a - c) = d - b.
Step 5: Divide both sides by (a - c): x = (d - b)/(a - c). This only works if a and c are not equal, because we cannot divide by zero.
Check with numbers: Let a = 3, b = 1, c = 1, d = 9. Then x = (9 - 1)/(3 - 1) = 4. Original: 3(4) + 1 = 13 and 1(4) + 9 = 13. They match.
Tips and common mistakes
- Treat every letter except the one you are solving for as a number.
- Undo operations in reverse order, like peeling an onion from the outside in.
- If your letter is in more than one term, you have to factor it out before you divide.
- If you divide by an expression, remember it cannot equal zero.
- If there are fractions, multiply both sides by the common denominator first.
- To check your work, put in simple numbers for all the letters. The original and your new formula should give the same answer.
- Do not simplify too early. It can hide the common factor you need.