Lesson 5 · from §80 of the 1910 text
Graphing Linear Equations
Lines parallel to the axes
The equation x = 4 is the same as x + 0y = 4. It is satisfied by x = 4 and any value of y: (4, 3), (4, 6), (4, 0), (4, -2), and so on. Its graph is a line parallel to the y-axis, 4 units to the right of it. Likewise y = -2 is a line parallel to the x-axis, 2 units below it.
The figure shows 3x + 4y = 12 (crossing the axes at (4, 0) and (0, 3)), x = 4, y = -2, and x - 2y = 0 (through the origin).
The constant term
In a linear equation, the constant term is the term with no variable: in 3x + 4y = 12 it is 12, and in ax + by = c it is c.
- If the constant term is zero, then x = 0 gives y = 0, so the line goes through the origin.
- If y is missing (its coefficient is zero), as in x = 4, the line is parallel to the y-axis. If x is missing, as in y = -2, it is parallel to the x-axis.
For the exercises, graph each line on paper, then answer the questions about where it crosses the axes.
Try one:
Quick check
Where does the graph of 2x - 3y = 12 cross the x-axis? the y-axis?
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Graph each equation on paper, then say where the line crosses each axis. Type none if it never crosses that axis, or any if the line is that axis.
x + y = 6
x - y = 5
x + 2 y = 8
3 x + 4 y = 12
4 x - 3 y = 12
2 x + 4 y = 9
x - 2 y = 0
3 x - y = 0
x = 4
x = -6
y = 5
y = -2
x = 0
y = 0
x = \pm 3
If a point is on a line, do the values of its x- and its y-distances satisfy the equation of the line?
If the values of the x- and the y-distances of a point satisfy the equation of a line, is the point located on the graph of the equation?
Is the point (3, 4) on the line whose equation is 3x - 4y = 12? Is (0, 4)? Is (4, 0)?
Can you determine without reference to the graph itself if the point (2, 6) is on any of the graphs of the equations in Exercises 1–9 above? If so, on which ones? Type the exercise number(s).
Which of the graphs of the equations in Exercises 1–14 pass through the origin? Type the exercise numbers, separated by commas.
What is the value of the constant term in the equations whose graphs pass through the origin? Is it ever that value in equations whose graphs do not pass through the origin?
Can you tell, then, by looking at a linear equation whether its graph goes through the origin or not?
Can you tell from the equation when a line is parallel to the x-axis? the y-axis? Say which letter's coefficient is zero in each case: type x or y.