← Pre-Algebra

Unit 5: Solving Equations

The balance scale that keeps both sides equal, and every move — one step to many, one variable to two — that uncovers what x is.

7 topics · 27 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Balance Scale Model of an Equation

  1. How it worksHow does doing the exact same thing to both sides of a balance scale keep it from tipping?
  2. Common mistakeWhy can't you add a number to just one side of an equation and expect it to still be true?
  3. Going furtherWhat happens to the balance scale picture when the unknown weight is hidden inside a box on both sides?
Your turn1 of 2
An equation x plus 7 equals 23 is balanced like a scale. If you add 8 to the left side, do you have to add 8 to the right side as well for the equation to stay true?

One-Step Equations

  1. How it worksHow does undoing the one operation in an equation isolate the variable?
  2. Worked exampleSolve x plus 9 equals 15 for x.
  3. Worked exampleSolve 4 times x equals negative 28 for x.
  4. Common mistakeWhy doesn't solving x minus 3 equals 7 mean subtracting 3 from both sides?
Your turn1 of 2
Ben had some trading cards, then found 14 more, ending up with 46 cards in total. Solve x plus 14 equals 46 to find how many cards, x, he started with.

Two-Step Equations

  1. How it worksHow does undoing addition or subtraction first, then undoing multiplication or division, peel an equation down to the variable?
  2. Worked exampleSolve 3x plus 5 equals 20 for x.
  3. Going furtherWhat changes about the two-step process when the equation starts with a fraction like x over 4 plus 3 equals 9?
  4. Common mistakeWhy doesn't dividing first instead of subtracting first change the final answer, even though it often makes the work messier?
Your turn1 of 2
A taxi ride costs 13 dollars just to start, plus 7 dollars per mile, x. The total came to 76 dollars. Solve 7x plus 13 equals 76 to find how many miles, x, were driven.

Multi-Step Equations

  1. How it worksHow do you distribute and combine like terms first, then solve the simpler two-step equation that's left?
  2. Worked exampleSolve 2 times the quantity x plus 3, minus 4x, equals 10 for x.
  3. Worked exampleSolve 5x minus 3 plus 2x equals 25 for x.
  4. Common mistakeWhy can't you undo an operation on x before combining the like terms sitting on the same side?
Your turn1 of 2
A moving company charges 6 dollars for every box, where each box holds x plus 6 items, then subtracts a discount of 3x dollars. If the total charge is 12 dollars, solve 6 times the quantity x plus 6, minus 3x, equals 12 for x.

Equations with Variables on Both Sides

  1. How it worksHow does moving all the variable terms to one side turn a both-sides equation into an ordinary two-step equation?
  2. Worked exampleSolve 5x plus 2 equals 2x plus 20 for x.
  3. Common mistakeWhy doesn't it matter whether you move the variable terms to the left side or the right side first?
  4. Going furtherWhat happens when you move the x terms and end up with the same number on both sides of the equation?
Your turn1 of 2
Gym A charges 8 dollars per month, x, plus a 19 dollar signup fee. Gym B charges 3 dollars per month plus a 17 dollar signup fee. Solve 8x plus 19 equals 3x plus 17 to find the number of months, x, where both gyms cost the same.

Equations with No Solution or Infinitely Many Solutions

  1. How it worksHow does an equation collapsing to a true statement mean every number works, while collapsing to a false statement means none do?
  2. Worked exampleSolve 2 times the quantity x plus 3 equals 2x plus 6 and explain what the result means.
  3. Worked exampleSolve 3x plus 4 equals 3x minus 1 and explain what the result means.
  4. Common mistakeWhy doesn't an equation like 4x equals 4x mean x has to equal 4?
  5. Ties togetherHow does the balance scale picture explain why some equations balance no matter what x is, and others never balance at all?
Your turn1 of 2
Solving an equation, a student simplifies both sides down to 14 equals 14 (the same number on each side, with the variable gone). What does this mean?

Writing and Solving an Equation from a Word Problem

  1. How it worksHow do you translate a word problem's setup fee and per-unit cost into a two-step equation?
  2. Worked exampleFind how many gigabytes were used, by writing and solving an equation, if a phone bill of 45 dollars comes from a 20 dollar base fee plus 5 dollars per gigabyte.
  3. Ties togetherHow does writing an equation from a word problem use the same translation skills as writing an expression from words?
Your turn1 of 2
Nina's phone bill costs 20 dollars plus 6 dollars per gigabyte used. If her total bill is 143 dollars, how many gigabytes did she use?