Given that the linear equations 2x + 3y - 8 = 0 write another linear equation whose geometrical representation of the pair so formed is: 1) intersecting lines, 2) parallel lines, 3... Given that the linear equations 2x + 3y - 8 = 0 write another linear equation whose geometrical representation of the pair so formed is: 1) intersecting lines, 2) parallel lines, 3) coincident lines.
Understand the Problem
The question asks for the formulation of different linear equations based on the given equation 2x + 3y - 8 = 0, to create three types of relationships between the lines: intersecting, parallel, and coincident. We will determine the parameters that will allow us to construct these equations accordingly.
Answer
- Intersecting Line: \(x - y = 0\) - Parallel Line: \(2x + 3y - 3 = 0\) - Coincident Line: \(4x + 6y - 16 = 0\)
Answer for screen readers
The equations for the different types of relationships are:
- Intersecting Line: (x - y = 0)
- Parallel Line: (2x + 3y - 3 = 0)
- Coincident Line: (4x + 6y - 16 = 0)
Steps to Solve
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Understanding the given equation The initial equation given is (2x + 3y - 8 = 0). We need to manipulate this equation to determine the slope and y-intercept.
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Rearranging the equation to slope-intercept form To find the slope (m) and y-intercept (b), we rearrange the equation into the form (y = mx + b):
$$ 3y = -2x + 8 $$
Dividing by 3:
$$ y = -\frac{2}{3}x + \frac{8}{3} $$
Here, the slope (m) is (-\frac{2}{3}).
- Formulating the parallel line equation Parallel lines have the same slope. Thus, we can choose any y-intercept (b) to create a parallel line. For example, let's use (b = 1):
$$ y = -\frac{2}{3}x + 1 $$
Rearranging back to standard form gives us the equation (2x + 3y - 3 = 0).
- Formulating the intersecting line equation To create a line that intersects, we need a different slope. Choosing a slope of (1) and a y-intercept of (0), we get:
$$ y = 1x + 0 $$
Rearranging gives the equation (x - y + 0 = 0).
- Formulating the coincident line equation A coincident line has identical coefficients. We can multiply the original equation by any non-zero constant, say (k = 2):
$$ 2(2x + 3y - 8) = 0 $$
This simplifies to:
$$ 4x + 6y - 16 = 0 $$
The equations for the different types of relationships are:
- Intersecting Line: (x - y = 0)
- Parallel Line: (2x + 3y - 3 = 0)
- Coincident Line: (4x + 6y - 16 = 0)
More Information
These forms of linear equations showcase important properties in geometry. The parallel line maintains the same slope as the original, whereas the intersecting line changes slope leading to a single point of intersection with the original line. The coincident line is simply a scaled version of the initial equation.
Tips
- Confusing parallel lines with intersecting lines by incorrectly keeping the same slope but changing the y-intercept.
- Forgetting to rearrange the angles properly into standard or slope-intercept form when formulating the equations.
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