Standard form of a line

Why this exists

Zainab’s class is raising money for a trip. They sell tins of shortbread at 5 dollars each and T-shirts at 8 dollars each, and they want to raise exactly 120 dollars. Call the number of tins sold x and the number of T-shirts y. The tins bring in 5x dollars, the shirts 8y, and together they make 120:

The story hands you this equation directly: each quantity times its own price on the left, the total alone on the right.

Two questions answer themselves. Only shortbread means y = 0, so 5x = 120 and x = 24 tins. Only T-shirts means x = 0, so 8y = 120 and y = 15 shirts. Those are the two points where the line meets the axes.

The same line in slope-intercept form is \displaystyle y = -\frac{5}{8}x + 15: correct, but it brings in a fraction and reads as though shirts depend on tins, which the story never said. Slope-intercept form suits a quantity that starts somewhere and changes at a steady rate; when two quantities share a fixed total, the form above is the natural one, and lines move freely between it, slope-intercept form and point-slope form.

Definition

A linear equation in x and y is in standard form when it is written as

where A, B and C are numbers and A and B are not both zero. Both variable terms are on the left, each times its own constant, and the constant stands alone on the right.

On top of this there is a convention, an agreement about which of many correct equations to write, not a rule of mathematics: A, B and C are integers with no common factor bigger than 1, and A is not negative (when A = 0, B is positive). The equations 10x + 16y = 240, 5x + 8y = 120 and -5x - 8y = -120 all have the form Ax + By = C and all describe the fundraiser line; the convention picks 5x + 8y = 120, so that two people writing the same line write the same equation. Some books use the name for any Ax + By = C.

The x-intercept of a line is the point where it crosses the x-axis, so its y-coordinate is 0. The y-intercept is where it crosses the y-axis, so its x-coordinate is 0.

Intuition

In standard form each variable carries its own weight and the weighted total is fixed, so moving along the line trades one quantity for the other. In the fundraiser, 5 T-shirts and 8 tins both bring in 40 dollars. To stay on 120 dollars, selling 5 fewer shirts must be made up by 8 more tins: 5 down for every 8 across, a slope of \displaystyle -\frac{5}{8}. The slope is hiding in the coefficients. The class sells whole tins and whole shirts, so only the four dots are sales they can make; the dashed line shows the pattern the dots follow.

The intercepts are quick because each comes from making a term vanish: y = 0 leaves the one-step equation Ax = C, and x = 0 leaves By = C. Two points fix a line, and a third point from the equation is a cheap check: if the three do not line up, one is wrong.

Results

Worked examples

Boundaries

• A and B are not both zero. 0x + 0y = C is not a line: if C = 0 every point makes it true, and if C \neq 0 none does.
• The slope \displaystyle -\frac{A}{B} needs B \neq 0. When B = 0 the line is vertical and has no slope.
• Each intercept needs its own coefficient non-zero. A horizontal line off the x-axis has no x-intercept, and a vertical line off the y-axis has no y-intercept. When C = 0 and A, B are non-zero, both intercepts are the origin, so a second point must come from elsewhere.
• Integer coefficients need a rational slope and intercept. \displaystyle y = \sqrt{2}\,x can be written \displaystyle \sqrt{2}\,x - y = 0, but not with integers: the slope would then be \displaystyle -\frac{A}{B}, a fraction of integers, and \displaystyle \sqrt{2} is not one.
• A story uses only part of its line. Amounts cannot be negative, and counted things (tins, goals) must be whole.

Where this goes next

Standard form is the natural layout for two equations at once: with x terms in one column and y terms in the next, Solving systems by elimination adds a multiple of one equation to the other so that a variable drops out. And budgets are rarely spent to the last cent: “at most 120 dollars” turns the fundraiser line into 5x + 8y \le 120, the edge of a region, as in Graphing a linear inequality in two variables, drawn from its intercepts just as here. Modeling with linear relationships uses every form, choosing whichever matches the story.

Exercises

Intercepts and graphing

1. Find both intercepts of 2x + 5y = 20.
2. Find both intercepts of x - 6y = 18.
3. Find both intercepts of 4x - 3y = -18.
4. Find both intercepts of 6x + 4y = 9.
5. Find the intercepts of 2x - 3y = 12, then find a third point with x = 3 to check before drawing the line.
6. Find the intercepts of 4x + 7y = 0. Explain why they are not enough to draw the line, and find another point on it with x = 7.

Moving between forms

7. Write y = 2x - 7 in standard form.
8. Write \displaystyle y = -\frac{4}{5}x + 3 in standard form.
9. Write \displaystyle y = \frac{1}{6}x - \frac{3}{4} in standard form.
10. Write \displaystyle y + 2 = -\frac{4}{3}(x - 1) in standard form.
11. Find the line through (1, 6) and (4, -3) and write it in standard form.
12. Write 9x - 3y = 6 in slope-intercept form, and then in standard form following the convention.
13. Use \displaystyle -\frac{A}{B} and \displaystyle \frac{C}{B} to give the slope and the y-intercept of 8x - 6y = 5, then check by solving for y.

Horizontal and vertical lines

14. Say whether each line is horizontal or vertical, and give every intercept it has: (a) 5y = 15; (b) 4x = -10.
15. Write each line in standard form: (a) the line through (-2, 6) and (-2, 1); (b) the line through (5, -3) and (-1, -3).

In context

16. Bettina’s club sells calendars at 6 dollars and keyrings at 4 dollars and wants to raise exactly 96 dollars. Let x be calendars and y keyrings. (a) Write the equation in standard form. (b) Find the intercepts and say what each means. (c) If they sell 10 calendars, how many keyrings do they need to sell?
17. In Aldric’s game a header scores 2 points and a long shot scores 5. His team scores 21 points with x headers and y long shots. Write the equation and list every possible combination of goals.

Reasoning

18. Show that 2x + 3y = 6, 4x + 6y = 12 and -2x - 3y = -6 all describe the same line. Which one does the convention pick, and why is a convention useful?
19. Zainab claims that keeping A and B in Ax + By = C but changing C gives a parallel line, and that doubling C doubles the distance of each intercept from the origin. Explain whether she is right, and name the case where doubling C gives no new line.