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 think about an equation with two letters, I think about shopping with a fixed amount of money. Say I have 6,000 RWF and I buy mandazi and tea. Each mandazi costs 200 and each tea costs 500. If I buy more mandazi, I must buy less tea. There is not one answer, there are many pairs that use the whole 6,000. Each pair is a point on a line. That is a linear equation in two variables. The equation does not give me one number. It gives me a family of pairs that all fit. On the SAT I ask what the line looks like and which pairs fit it. Then I read the question again.
These questions can look long, but they only test a few ideas. Slope, intercepts, and plugging in a point. Learn those three and you will answer most of them.
What is inside this part
You decide whether a pair is a solution, or find a missing coordinate.
Substitute a point · Find the missing y
You read or find the slope, the y-intercept and the x-intercept.
Standard form to slope · Intercept points and areas
You make the equation from two points, a slope, or a story.
Two points · Point and slope
You use slopes to relate two lines.
Same slope · Slopes that multiply to −1
How to know a question is from this part
An equation with x and y, or a story with two kinds of items and a total.
- Both x and y in the equation
- "Which of the following points lies on the line"
- "What is the slope" or "y-intercept" or "x-intercept"
- "Parallel" or "perpendicular" to a given line
- A story with two items, two prices and one total
The lesson
1. What is it?
A linear equation in two variables has two letters, usually x and y, each to the power 1. Examples: 2x + 3y = 12 and y = 3x − 1.
A solution is a pair (x, y) that makes the equation true. For 2x + 3y = 12, the pair (3, 2) is a solution because 2(3) + 3(2) = 12. There are infinitely many solutions, and together they make a straight line on a graph.
2. Three ways to write a line
Slope-intercept form: y = mx + b. Here m is the slope and b is where the line crosses the y axis.
Standard form: Ax + By = C. This is common on the SAT, especially in word problems about totals.
To change standard form to slope-intercept form, solve for y. For 4x − 2y = 10, subtract 4x to get −2y = −4x + 10, then divide by −2 to get y = 2x − 5. The slope is 2.
What is the slope of the line 4x − 2y = 10?
Show the steps
- Solve for y: −2y = −4x + 10.
- Divide every term by −2: y = 2x − 5.
- The number with x is the slope.
2
3. Checking a point
To test if a point is on a line, put its x and y into the equation. If the sentence is true, the point is on the line.
Example: Is (2, 4) on 3x + y = 10? 3(2) + 4 = 10. Yes.
4. Intercepts
The y-intercept is where x = 0. The x-intercept is where y = 0. To find an intercept, set the other letter to 0 and solve.
For 5x + 3y = 30: set x = 0 to get y = 10. Set y = 0 to get x = 6.
The line and the two axes make a right triangle. Its area is ½ × 6 × 10 = 30.
5. Writing the equation of a line
With two points, first find the slope. Then use y = mx + b with one point to find b.
Example: points (1, 2) and (4, 11). The slope is (11 − 2) ÷ (4 − 1) = 3. Put in (1, 2): 2 = 3 + b, so b = −1. The line is y = 3x − 1, which is the same as 3x − y = 1.
6. Parallel and perpendicular lines
Parallel lines have the same slope. Perpendicular lines have slopes that multiply to −1, for example 2 and −½.
If a line is given in standard form Ax + By = C, its slope is −A ÷ B. That shortcut saves time.
7. Word problems
In a story, each letter stands for a count of something. The equation is usually total cost or total amount. Price × amount of the first thing + price × amount of the second thing = total.
If one of the counts is given, put it in and solve for the other.
Pens cost 200 RWF and notebooks cost 500 RWF. Jean spent 4,600 RWF and bought 6 notebooks. How many pens did he buy?
Show the steps
- Write 200p + 500n = 4600.
- Put n = 6: 200p + 3000 = 4600.
- 200p = 1600, so p = 8.
8 pens
8. Common traps
Trap 1: forgetting to divide the sign when you divide by a negative.
Trap 2: reading the slope straight from standard form. In 4x − 2y = 10 the slope is not 4.
Trap 3: forgetting the ½ in triangle area.
Trap 4: mixing up the x-intercept and y-intercept.
See it
| Form | Looks like | Slope | y-intercept |
|---|---|---|---|
| Slope-intercept | y = mx + b | m | b |
| Standard | Ax + By = C | −A ÷ B | C ÷ B |
| Intercept method | Set x = 0 or y = 0 | From the two points | Set x = 0 |
- 1Start with Ax + By = C
- 2Subtract Ax from both sides
- 3Divide every term by B
- 4Read y = mx + b
- 5The slope is the number with x
The axes and the line make a right triangle with area ½ × 6 × 10 = 30.
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