Lesson 4 · from §80 of the 1910 text
The Graph of an Equation
Finding pairs of values
To graph an equation, find several pairs of values that satisfy it. Give x a value and solve for y, or give y a value and solve for x.
Example. For 3x + 2y = 6: if x = 0, then 2y = 6 and y = 3. If x = 4, then 12 + 2y = 6 and y = -3. If y = 0, then x = 2. Plotting (0, 3), (4, -3), and (2, 0), we find them in a straight line.
The graph of an equation of the first degree in x and y (a linear equation) is always a straight line. Since two points fix a line, it is enough to plot two points and draw the line through them. The handiest two are usually where the line crosses the axes: put x = 0 to find one and y = 0 to find the other.
Example. Graph 2x + 5y = 10. If x = 0, y = 2; if y = 0, x = 5. Plot (0, 2) and (5, 0) and draw the line.
If the two crossing points are very close together, choose two values of x far apart (such as 0 and 5) instead. If the line goes through the origin, (0, 0) gives only one point, so take a second point away from the axes.
Check: plot a third point whose coördinates satisfy the equation. If it lies on your line, the line is right.
The equation of a line is satisfied by the coördinates of any point on that line, and any point whose coördinates satisfy the equation is on the graph.
Try one:
Quick check
Is the point (10, -2) on the graph of 2x + 5y = 10?
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.
Find pairs of values of x and y which satisfy the equation x + 2y = 8, and plot them with ½ inch as the unit distance. Are the points in a straight line? Does x = 4, y = 4 satisfy this equation, and is the point (4, 4) on the graph?
Find pairs of values for x and y which satisfy each equation (have at least one negative value for x and one for y), then plot the points. Give the values asked for.
3 x + 2 y = 6
3 x - 4 y = 12
x + y = 0
x - y = 0
x = 3 y
y = 2 x