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Nonlinear equations in one variable and systems of equations in two variables: line and curve
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. Both sides equal, then one side zero
- Start:
- Read the two equations and see which letter is alone (usually y).
- Stop:
- Stop when you have a quadratic equal to 0.
- Skip:
- Skip graphing by hand and skip solving for both letters at once.
- Why:
- Questions 4, 5, 6 and 8 all begin by setting or substituting into one equation. That is the same first step every time.
Used in questions 4, 5, 6, 8
2. Read the last sentence first: x, y, a sum, or a constant?
- Start:
- Go to the last sentence.
- Stop:
- Stop reading when you know what the answer must be (a point, a value of x, a sum).
- Skip:
- Skip extra steps once you have what is asked.
- Why:
- In question 6, the answer is x + y, and in question 1 it is the greater solution. Solving more than that wastes time.
Used in questions 1, 6
3. Exactly one solution means discriminant zero
- Start:
- Find the words 'exactly one', 'tangent', or 'one point'.
- Stop:
- Stop when you have b² - 4ac = 0 and solved for the unknown.
- Skip:
- Skip solving for x.
- Why:
- Questions 5 and 8 are solved just by the discriminant. In question 8, the sign trick: the c² terms combine to -4c² + 400.
Used in questions 5, 8
4. Always check square-root answers
- Start:
- After squaring, solve normally.
- Stop:
- Stop after putting every answer back into the original equation.
- Skip:
- Skip the long algebra, but never skip the check. It takes 10 seconds.
- Why:
- In questions 2 and 7 the original equation is the judge. Question 7 has a fake answer, -1.
Used in questions 2, 7
5. Test the options in the equation
- Start:
- Take the answer options for an ordered pair or a value.
- Stop:
- Stop at the first option that fits both equations.
- Skip:
- Skip solving the whole system.
- Why:
- Question 4 can be solved by plugging each pair into both equations. Remember to test BOTH equations, because (4, 5) fits only one.
Used in question 4
6. Use Desmos to see the meeting points
- Start:
- Type both equations into Desmos.
- Stop:
- Stop when you can read the intersection points off the graph.
- Skip:
- Skip algebra when the options are neat numbers.
- Why:
- Questions 4 and 6 have whole-number crossing points that Desmos shows at once. For a circle, type x^2+y^2=25 exactly as written. For questions 5 and 8, slide a value of k or c to see the line touch the curve.
Used in questions 4, 5, 6, 8
7. Rational equation: write the forbidden values first
- Start:
- Find every bottom (denominator) and write what makes it zero.
- Stop:
- Stop after solving, and cross out any answer on the forbidden list.
- Skip:
- Skip checking answers that already passed the forbidden list.
- Why:
- In question 13, x = 3 comes out of the algebra but makes a bottom zero. It is exactly the trap option.
Used in question 13
8. Sum and product of the solutions
- Start:
- Write the equation as ax² + bx + c = 0 and note: sum = -b ÷ a, product = c ÷ a.
- Stop:
- Stop when you have two simple facts about the two solutions (sum and difference, or sum and product).
- Skip:
- Skip the quadratic formula.
- Why:
- Question 16 gives the difference, and the sum comes from the equation. The solutions 5 and 1 appear at once. This idea also checks question 1: 2 + 3 = 5.
Used in questions 1, 9, 16
9. On a graph, a solution must lie on both curves
- Start:
- Find where the curves cross, then test the option pairs in BOTH equations.
- Stop:
- Stop at the first pair that fits both.
- Skip:
- Skip any pair that is on only one graph.
- Why:
- Question 10 has wrong options that fit just the parabola or just the line. In question 15, A and B are the crossing points of the line and circle, so find them by substitution, then use the distance formula.
Used in questions 4, 10, 15