Linear equations in two variables
Algebra is about 35% of SAT Math (13β15 of 44 questions). Linear equations in two variables appear in both modules, often as a word problem with two kinds of items, or as a question about which line passes through given points.
Three ways to write a line
The same line can be written in different forms. Slope-intercept form, y = mx + b, shows the slope and the y-intercept at once. Standard form, ax + by = c, makes it easy to find both intercepts and fits stories with two kinds of items. Point-slope form, y β yβ = m(x β xβ), is the quickest way to write a line when you know one point and the slope. Moving between forms is only algebra.
| Form | Looks like | Best for |
|---|---|---|
| Slope-intercept | y = mx + b | reading the slope and y-intercept |
| Standard | ax + by = c | intercepts; stories with two items |
| Point-slope | y β yβ = m(x β xβ) | a point and a slope are known |
Reading a line in standard form
To find the intercepts of ax + by = c, put y = 0 for the x-intercept (x = c/a) and x = 0 for the y-intercept (y = c/b). To find the slope, solve for y: y = β(a/b)x + c/b, so the slope is βa/b. For 3x + 4y = 24: x-intercept 24/3 = 8, y-intercept 24/4 = 6, slope β3/4.
Writing an equation from two points
Follow the same steps every time: find the slope from the two points, put one point into y = mx + b to find b, then write the equation. Check by putting the other point into your equation.
Parallel and perpendicular lines
Two lines are parallel when their slopes are equal and their y-intercepts are different. Two lines are perpendicular when the slopes are negative reciprocals: flip the fraction and change the sign (the slopes multiply to β1). To compare lines in standard form, first find each slope with βa/b.
| Relationship | Slopes | Example |
|---|---|---|
| Parallel | equal | y = 3x + 1 and y = 3x β 5 |
| Perpendicular | negative reciprocals | y = 2x + 1 and y = βx/2 + 4 |
| Neither | anything else | y = 2x + 1 and y = 3x + 1 |
Stories with two kinds of items
Let x and y be the numbers of two kinds of items, with prices (or sizes) p and q. If the total is T, then px + qy = T. A coffee stand sells small cups for $3 and large cups for $5 and takes $120: 3s + 5l = 120. The intercepts have meaning: (40, 0) means 40 small cups and no large cups; (0, 24) means 24 large cups and no small cups. A point on the line is a combination that gives exactly $120. Only whole numbers make sense in the story.
- Slope: (14 β 5) Γ· (4 β 1) = 9 Γ· 3 = 3.
- Put (1, 5) into y = 3x + b: 5 = 3 + b, so b = 2.
- The equation is y = 3x + 2.
- Check (4, 14): 3(4) + 2 = 14 β
- Find the slope of m: solve 2x β 5y = 15 for y. β5y = β2x + 15, so y = (2/5)x β 3. The slope is 2/5.
- Parallel lines have the same slope: n has slope 2/5.
- Put (5, 4) into y = (2/5)x + b: 4 = 2 + b, so b = 2.
- The y-intercept of n is 2.
- The given slope is β3/4. The negative reciprocal is 4/3.
- Put (6, 1) into y = (4/3)x + b: 1 = 8 + b.
- So b = β7.
- Put r = 15: 8(15) + 6f = 240, so 120 + 6f = 240.
- 6f = 120, so f = 20.
- For (0, 40): r = 0 and f = 40, and 6(40) = 240 β. It means the club could spend all $240 on 40 bags of flour and no rice.
- Forgetting the minus sign in βa/bFor ax + by = c the slope is βa/b. For 3x + 4y = 24 the slope is β3/4, not 3/4. When in doubt, solve for y.
- Perpendicular slope: flipping but not changing the signThe perpendicular slope to 2/3 is β3/2. Do both: turn the fraction upside down and change the sign.
- Thinking parallel lines share the interceptParallel lines share the slope. If they also shared the y-intercept they would be the same line.
- Dividing only one term when changing formTo solve 6x β 2y = 10 for y, every term is divided by β2: y = 3x β 5. Write each term.
- Swapping x and y when you substitute a pointA point (a, b) means x = a and y = b. Write βx =β and βy =β next to the numbers if you often swap them.
- Misreading what an intercept means in a storyThe x-intercept is where y = 0: the first variableβs amount when none of the second item is used. Say it in the words of the story.
Desmos accepts any form of a line: type 3x + 4y = 24 exactly as written and it draws the line. Click the line to see its intercepts. To test answer choices, type each choice on its own line and check which one passes through the given points, which you can type as (1, 5) and (4, 14). For parallel and perpendicular questions, type both lines and look at the picture; but check the slopes by hand, because a graph can look square when it is not.
- Type 2x β 5y = 15
- Click the line to see where it meets the axes: (7.5, 0) and (0, β3)
- Type the point (5, 4) and a second line with the same slope that passes through it
- Type each answer choice and keep the one that goes through all the points
Use Desmos when the question gives points and answer choices: plotting is faster than solving. For writing the equation of a line through two nice points, the slope formula is usually faster.
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