Question 2.5: PLOT THE LINE ax + by + d = 0 The equation of a line is give......
PLOT THE LINE ax + by + d = 0
The equation of a line is given as 4x + 2y − 8 = 0.
(a) Write the equation of the line in the form y = mx + c, hence write down the slope and y-intercept. Give a verbal description of slope and intercept.
(b) Which variable should be plotted on the vertical axis and which on the horizontal axis when the equation of the line is written in the form y = mx + c?
(c) Plot the line by (i) calculating a table of points (method A) and (ii) calculating the intercepts (method B).
(a) Rearrange the equation of the line with y as the only variable on the LHS:
4x + 2y − 8 = 0
2y = 8 − 4x bringing all non-y-terms to the RHS
\frac{2y}{2} = \frac{8}{2} – \frac{4x}{2} dividing everything on both sides by 2
y = 4 − 2x simplifying
From the rearranged equation we can read off useful information before going ahead to plot the line. Intercept: the line cuts the y-axis at y = 4. So when x = 0, y = 4. Slope: the line has slope m=−2, i.e., the line is falling from left to right, dropping by 2 units for each horizontal increase of 1 unit.
(b) When the equation is written in the form y = mx + c with the variable by itself on one side of the equation, y is usually plotted on the vertical axis and the other variable, x, is plotted on the horizontal axis.
(c)
| (i) Calculate a table of points | (ii) Calculate the intercepts |
| Step 1: y = −2x + 4: see part (a) | Step 1: y = −2x + 4: see part (a) |
| Step 2: From the equation of the line
y=−2x + 4, calculate y for several x-values, such as: x = 0, y=−2(0) + 4 = 4: the point is (0, 4) x = 1, y=−2(1) + 4 = 2: the point is (1, 2) x = 2, y=−2(2) + 4 = 0: the point is (2, 0) x = 3, y=−2(3) + 4=−2: the point is (3, −2) |
Step 2: The equation is y=−2x + 4 The line cuts the y-axis at y = 4
Hence the vertical intercept is (0, 4) The line cuts the x-axis at x=−c/m= −4/(−2) = 2 Hence the horizontal intercept is (2, 0)^∗ |
| Step 3: Plot these points and draw the line through them (Figure 2.12) | Step 3: Plot the intercepts, join them and you have the graph of the line (Figure 2.12) |
| ^∗ Alternatively, solve for the x-intercept by using the fact that y = 0 at the point where the line cuts the x-axis | |
