← Algebra I

Unit 2: Solving Linear Equations

Solving an equation is keeping a scale balanced -- every move you make to one side you make to the other, until x stands alone.

6 topics · 25 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.

What It Means to Solve an Equation

  1. How it worksHow does thinking of an equation as a balanced scale explain why both sides change together?
  2. Worked exampleShow me why x = 6 makes the equation 2x - 4 = 8 true.
  3. Common mistakeWhy doesn't adding a number to only one side of an equation keep it balanced?
  4. Going furtherWhat happens to the balance of an equation if you multiply just one side by 2?
  5. Worked exampleShow me why x = -3 makes 5x + 12 = -3 true.
Your turn1 of 2
To solve 7x + 6 = 18 you subtract 6 from the left side. What must you also do to keep the equation balanced?

One-Step Equations

  1. How it worksHow do you pick the opposite operation to undo whatever is being done to x?
  2. Worked exampleFind the value of x that solves x + 9 = 15.
  3. Worked exampleFind the value of x that solves 4x = 28.
  4. Common mistakeWhy doesn't dividing just the 4 by 4 in 4x = 28 leave the rest of the equation unchanged?
Your turn1 of 2
Find the value of x that solves x + 14 = 14.

Two-Step Equations

  1. How it worksHow does peeling off operations in reverse order isolate x in a two-step equation?
  2. Worked exampleFind the value of x that solves 3x - 7 = 14.
  3. Worked exampleFind the value of x that solves -2x + 9 = 1.
  4. Common mistakeWhy can't you divide only the 3x term by 3 and leave the rest of 3x - 7 = 14 untouched?
Your turn1 of 2
Find the value of x that solves 7x - 10 = 27.

Equations with Variables on Both Sides

  1. How it worksHow does moving all the x-terms to one side turn a two-sided equation into a one-step equation?
  2. Worked exampleFind the value of x that solves 5x - 3 = 2x + 9.
  3. Worked exampleFind the value of x that solves 7 - 2x = 3x + 22.
  4. Common mistakeWhy doesn't subtracting 2x from one side mean you skip subtracting it from the other side too?
Your turn1 of 2
Find the value of x that solves 8x - 6 = 4x + 5.

Equations That Need Distributing First

  1. How it worksHow does distributing across parentheses turn a hidden two-sided equation into a plain one?
  2. Worked exampleWalk me through solving 3 times the quantity x minus 4, equals 2x plus 7.
  3. Worked exampleFind the value of x that solves 2 times the quantity x plus 5, equals 3 times the quantity x minus 1.
  4. Common mistakeWhy can't you subtract the 4 from inside 3 times the quantity x minus 4, before the 3 has been distributed?
Your turn1 of 2
Find the value of x that solves 7 times the quantity x minus 6, equals 3x + 6.

Equations with No Solution or Infinitely Many

  1. How it worksHow does an equation collapsing to a true statement like 6 = 6 mean every value of x works?
  2. Worked exampleSolve 2 times the quantity x plus 3, equals 2x plus 6, and explain what the result means.
  3. Worked exampleSolve 4x + 1 = 4x - 5 and explain what the result means.
  4. Common mistakeWhy doesn't x canceling out completely mean you made a mistake?
Your turn1 of 2
Solving 7x + 6 = 7x + 19, the x-terms cancel out. Given that 6 does not equal 19, what is the solution set?