What this lesson covers
- The three cases: different slopes, parallel lines, and one line drawn twice
- The money beat: 3x + 2y = 12 vs 6x + 4y = 30, and why 0 = 6 means no solution
- The twist: find k so 4x + 6y = 10 and 6x + ky = 15 have infinitely many solutions
- Practice: kx + 3y = 9 and 2x + y = 4 have no solution, find k
- The three traps that eat most wrong answers
Questions worked in the video
- 0:58How many solutions does 3x + 2y = 12 and 6x + 4y = 30 have?
- 1:40For what k do 4x + 6y = 10 and 6x + ky = 15 have infinitely many solutions?
- 2:12kx + 3y = 9 and 2x + y = 4 have no solution. What is k?
Chapters
- 0:00Intro
- 0:04Two equations, nothing to solve
- 0:24The three cases
- 0:58The money beat: 0 = 6
- 1:40The twist: solve for k
- 2:12Practice: find k for no solution
- 2:50Three traps
- 3:21Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:04Two equations, nothing to solve
Welcome to SAT Climb. Two equations, and nothing to solve. The SAT wants to know how many solutions the system has. One, none, or infinitely many. And you can answer that just by looking at the coefficients.
0:24The three cases
Two linear equations are two lines. Everything depends on how the lines sit. Different slopes, and the lines cross once. That is exactly one solution. Same slope but different intercepts, and the lines run parallel. They never touch. No solution. Same slope and the same intercept, and they are the same line drawn twice. Every point works, so there are infinitely many solutions. The real question is never how many. It is this. Are the slopes the same, and if they are, are the lines the same.
0:58The money beat: 0 = 6
Here is the classic. 3x + 2y = 12, and 6x + 4y = 30. Do not solve. Compare. Multiply the first equation by 2 and the left side becomes exactly 6x + 4y, the same left side as the second equation. But the right side becomes 24, while the second equation says 30. Same left side, two different right sides. That is impossible. Push the elimination through and the variables cancel, leaving 0 = 6, a false statement. Parallel lines. No solution.
1:40The twist: solve for k
Now the version the test actually loves, with a missing constant. 4x + 6y = 10, and 6x + ky = 15. For infinitely many solutions, every ratio has to match. 6 over 4 is 3/2. 15 over 10 is 3/2. So k over 6 must also be 3/2, which makes k equal 9. Check it. Multiply the first equation by 3/2 and you get 6x + 9y = 15. Identical equations. One line drawn twice, so infinitely many solutions.
2:12Practice: find k for no solution
Your turn. kx + 3y = 9, and 2x + y = 4. The system has no solution. What is k? Pause and think. Here is the read. No solution means the slopes match but the lines are different. The y coefficients are 3 and 1, so the second equation scales up by 3, giving 6x + 3y = 12. Match the x terms and k must be 6. Now check the constants. 12 is not 9, so the lines really are parallel. k equals 6. Answer C.
2:50Three traps
Three traps. One, seeing the same slope and calling it no solution. Same slope can also mean the same line, which is infinitely many. Check the constant before you decide. Two, comparing only the x coefficients. All three ratios have to match, x, y, and the number on the right. Three, panicking when the variables vanish. 0 = 6 is not a mistake, it is the answer. It means no solution. And 0 = 0 means infinitely many.
3:21Recap
Systems, solved. Different slopes, one solution. Same slope with a different constant, none. Same everything, infinitely many. Compare the ratios before you solve anything. Start free at satclimb.com.
Read the written version: the Systems of two linear equations strategy guide, then try 25 hard Systems of two linear equations questions with full explanations. Both are free, no account needed.