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Unit 5: Systems of Equations

Two lines drawn on the same plane -- their intersection point is the one pair of numbers that satisfies both equations at once.

6 topics · 22 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 a System's Solution Means

  1. How it worksHow does checking a point in both equations confirm it's the solution to a system?
  2. Worked exampleFind where y = 2x + 1 and y = -x + 7 cross.
  3. Common mistakeWhy doesn't a system of equations have to have exactly one solution?
  4. Going furtherWhat happens to a system's solution when one of its two equations is multiplied by a constant?
Your turn1 of 2
A point (7, 6) is checked against a system. It makes the first equation true but makes the second equation false. Is it the system's solution?

Solving Systems by Graphing

  1. How it worksHow do you graph two equations on the same axes to spot their intersection?
  2. Worked exampleSolve the system y = x + 2 and y = -2x + 8 by graphing both lines.
  3. Common mistakeWhy doesn't graphing give you an exact answer when the intersection falls between grid lines?
Your turn1 of 2
Solve the system y = 7x + 6 and y = -1x + 18 by finding where they cross -- give the y-coordinate.

Solving Systems by Substitution

  1. How it worksHow does solving one equation for a variable let you substitute it into the other equation?
  2. Worked exampleSolve the system y = 3x - 1 and 2x + y = 9 using substitution.
  3. Worked exampleSolve the system x = 2y + 5 and 3x - y = 20 using substitution.
  4. Common mistakeWhy can't you substitute an expression for y into the very same equation it came from?
Your turn1 of 2
Solve the system y = 4x - 6 and 4x + y = 18 using substitution. Find x.

Solving Systems by Elimination

  1. How it worksHow do you multiply one equation by a constant so a variable's coefficients cancel when added?
  2. Worked exampleSolve the system 3x + 2y = 16 and 5x - 2y = 8 using elimination.
  3. Worked exampleSolve the system 2x + 3y = 7 and 4x - y = 19 using elimination.
  4. Common mistakeWhy do you have to multiply an entire equation by a number, not just the term you want to cancel?
Your turn1 of 2
Solve the system 4x + 5y = 29 and 4x - 5y = 2 using elimination. Find x.

Systems with No Solution or Infinitely Many

  1. How it worksHow does a system's equations turning into the exact same line mean infinitely many solutions?
  2. Worked exampleSolve the system y = 2x + 5 and y = 2x - 1 and explain what the result means.
  3. Worked exampleSolve the system y = 3x + 4 and 2y = 6x + 8 and explain what the result means.
  4. Common mistakeWhy doesn't a variable canceling out to a false statement mean you made an arithmetic error?
Your turn1 of 2
A system is y = 7x + 6 and y = 7x + 19, where 6 does not equal 19. What does the system's solution set look like?

Systems of Equations Word Problems

  1. How it worksHow do you translate two separate pieces of a word problem into two equations in the same variables?
  2. Worked exampleFind how many adult and child tickets were sold if 200 tickets sold for 1900 dollars total, adults at 12 dollars and children at 7 dollars.
  3. Going furtherWhat happens to a ticket-sales system's solution if the total revenue changes but the ticket count stays the same?
Your turn1 of 2
A total of 114 tickets were sold for a play. Adult tickets cost 12 dollars and child tickets cost 7 dollars, and the total revenue is r dollars. Write the number of adult tickets sold as a function of r.