On the test
Algebra is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder within the module, so Algebra shows up from the first question to the last.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I see two equations that share the same letters, I think about two friends splitting a taxi bill. One friend says, "Together we paid 10,000." The other says, "You paid 2,000 more than me." Each sentence alone is not enough, because many pairs of payments add up to 10,000. But both sentences together give only one pair. That is the idea of a system: two facts, one answer that fits both. On the SAT, I treat the two equations like the two sentences, and I look for the one pair that makes both true. I pick the easiest way to combine them, and I always read what the question wants at the end.
These questions seem hard because there are two equations, but the methods are fixed and you can learn them. After a few practice rounds, you will find them satisfying.
What is inside this part
You find x, y, or a combination like x + y.
Substitution · Elimination
You build two equations from a story, usually a count and a total.
Tickets and prices · Coins and money
You decide, or choose a constant so that, the system has none, one or infinitely many solutions.
Parallel lines · Same line
The equations have fractions or decimals that you clear first.
Multiply by the bottom number
How to know a question is from this part
Two equations given together, or a story with two unknowns and two facts.
- Two equations with x and y listed together
- "The system of equations" or "the solution (x, y)"
- "How many solutions" or "no solution" or "infinitely many solutions"
- A story with a count and a total value
- The words "in total" and "altogether" with two kinds of items
The lesson
1. What is a system?
A system of equations is two or more equations that use the same letters. We want the pair of values that makes all the equations true at the same time.
On a graph, each equation is a line. The solution is the point where the two lines cross.
2. Method 1: Substitution
Use substitution when one equation already has a letter alone, like y = 2x. Replace that letter in the other equation.
Example: y = 2x and x + y = 15. Replace y with 2x in the second equation: x + 2x = 15, so 3x = 15 and x = 5. Then y = 2(5) = 10.
If y = 2x and x + y = 15, what is y?
Show the steps
- Substitute: x + 2x = 15.
- 3x = 15, so x = 5.
- y = 2 × 5 = 10.
10
3. Method 2: Elimination
Use elimination when both equations are in the form Ax + By = C. Add or subtract the equations so that one letter disappears. Sometimes you must first multiply an equation to make the numbers match.
Example: x + y = 10 and x − y = 2. Add them: 2x = 12, so x = 6. Then y = 4.
Solve 3x + 2y = 16 and x − y = 2. What is x + y?
Show the steps
- From x − y = 2, x = y + 2.
- Put in the first: 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2.
- Then x = 4, so x + y = 6.
6
4. Word problems
Choose a letter for each unknown. Write one equation for the count and one for the money (or other total). Then solve.
Example: 40 tickets are sold. Adult tickets cost 5,000 RWF and child tickets cost 2,000 RWF. The total is 164,000 RWF. Let a be adults and c be children. a + c = 40 and 5000a + 2000c = 164000.
Use the ticket example. How many adult tickets were sold?
Show the steps
- Write c = 40 − a.
- 5a + 2(40 − a) = 164 (in thousands). That gives 3a + 80 = 164.
- 3a = 84, so a = 28.
28 adult tickets
5. How many solutions?
Two lines can cross once (one solution), be parallel (no solution), or be the same line (infinitely many solutions).
If the two equations have the same slope but different intercepts, they are parallel. For 2x + 3y = 7 and kx + 6y = 20, double the first one to get 4x + 6y = 14. For no solution, k = 4, and 14 is not 20.
If the equations are exactly the same line, there are infinitely many solutions. Multiply one equation by a number so that all three of x, y and the right-hand side match.
6. Fractions and decimals
If the equations have fractions, multiply every term by the common bottom number first to clear them. Then use substitution or elimination as before.
Example: x/2 + y/3 = 7 becomes 3x + 2y = 42 when you multiply by 6.
7. Common traps
Trap 1: solving for x but the question asks for something else, like x + y.
Trap 2: sign errors when subtracting equations. Subtract every term.
Trap 3: forgetting to check the second equation.
Trap 4: in no solution questions, forgetting that the right-hand side must be different.
See it
| The equations look like | Use | Why |
|---|---|---|
| y = ... or x = ... | Substitution | A letter is already alone |
| Ax + By = C and Dx + Ey = F | Elimination | Add or subtract to remove a letter |
| Fractions | Clear fractions, then either | Whole numbers are easier |
| Choices are numbers | Test the options | No algebra needed |
- Cross once: one solution
- Parallel: none
- Same line: infinitely many
- Different slopes
- Same slope, different numbers
- Same slope, same numbers
The solution of a system is the point where the two lines cross.
Videos that can help
Ask in the community. Tutors and other students answer.
Ask a question