A graph has the equation x + 2y + 4 = 0. a) Show that this is the equation of a straight line. b) Find the gradient of the line. c) Draw a graph of the line.
The Correct Answer and Explanation is:
a) Show that this is the equation of a straight line
The given equation is:
x + 2y + 4 = 0
To check if this is the equation of a straight line, rearrange it into the standard form of a straight line equation:
y = mx + c
Where:
- m is the gradient (slope) of the line
- c is the y-intercept
Starting with:
x + 2y + 4 = 0
Subtract x and 4 from both sides:
2y = -x – 4
Divide both sides by 2 to solve for y:
y = (-1/2)x – 2
This is now in the form y = mx + c, which is the standard equation of a straight line. Thus, the given equation represents a straight line.
b) Find the gradient of the line
From the rearranged equation:
y = (-1/2)x – 2
The coefficient of x is the gradient m, so:
Gradient (m) = -1/2
c) Draw a graph of the line
To plot the graph, identify at least two points on the line:
Find y-intercept
When x = 0:
y = (-1/2)(0) – 2 = -2
So the point is (0, -2)
Find x-intercept
Set y = 0:
0 = (-1/2)x – 2
Multiply both sides by 2:
0 = -x – 4
x = -4
So the point is (-4, 0)
Plotting Points
- First point: (0, -2)
- Second point: (-4, 0)
Draw a straight line passing through these points. The line slopes downwards from left to right because the gradient is negative.
Explanation
The given equation is linear because it can be rearranged to the form y = mx + c, with m = -1/2 and c = -2. This form confirms that the graph is a straight line. The gradient tells us the line slopes downward at a rate of one unit down for every two units to the right. The intercepts help in sketching the graph accurately.
