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Systems of two linear equations in two variables
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Add or subtract first
- Start:
- Both equations when the letters line up
- Stop:
- When one letter cancels
- Skip:
- Substitution if elimination is faster
- Why:
- In question 1, adding removes y at once.
Used in questions 1, 3
2. A letter alone: substitute
- Start:
- The equation with a letter alone
- Stop:
- After you replace and solve
- Skip:
- Elimination
- Why:
- Question 2 has y = 2x, so substitution is the quickest.
Used in question 2
3. Read the asked expression last
- Start:
- The last line of the question
- Stop:
- When you build x + y or a + 2b
- Skip:
- Finding each letter separately
- Why:
- Questions 3, 6 and 7 ask for a combination of letters, not one letter.
Used in questions 3, 6, 7
4. Word problem: count and total
- Start:
- The count and the money
- Stop:
- When you have two equations
- Skip:
- The story detail without numbers
- Why:
- Write a + c = count and price equation (question 4).
Used in question 4
5. No solution: scale, then compare
- Start:
- The numbers in front of x and y
- Stop:
- When the constants disagree
- Skip:
- Solving for x
- Why:
- Make the x and y numbers equal in both equations, then compare the right sides (question 5).
Used in question 5
6. Fractions out first
- Start:
- The equations with fractions
- Stop:
- After you multiply to clear them
- Skip:
- Fraction arithmetic
- Why:
- Multiplying by the bottom numbers makes whole-number equations (question 7).
Used in question 7
7. Desmos: graph both lines
- Start:
- The two equations
- Stop:
- When you click the crossing point
- Skip:
- Algebra when numbers are ugly
- Why:
- Type both equations in Desmos and click the point where they cross (questions 1, 3, 6, 7).
Used in questions 1, 3, 6, 7
8. Infinitely many: same line
- Start:
- The coefficients of the first equation
- Stop:
- When you multiply the first equation to match
- Skip:
- Solving the system
- Why:
- Find the multiplier from x, then multiply every term (question 8).
Used in question 8
9. Table rows are equations
- Start:
- Each row of the table
- Stop:
- When every row is written as an equation
- Skip:
- The row labels
- Why:
- Every row gives one equation with the same unknown prices. Subtract rows that share a number (questions 10, 12, 14).
Used in questions 10, 12, 14
10. Same line? Count the solutions
- Start:
- The two equations when one looks like a multiple of the other
- Stop:
- When you reach a true or false statement
- Skip:
- Solving for x and y
- Why:
- If substitution ends in a true statement like −10 = −10, there are infinitely many solutions. A false one means none (question 11).
Used in question 11
11. Meeting and chasing: add or subtract the speeds
- Start:
- The sentence that says toward each other or the same direction
- Stop:
- When you have a + b and a − b
- Skip:
- The names of the towns
- Why:
- Toward each other the speeds add; in the same direction you subtract (question 16).
Used in question 16