0GRAPHS:
All of the linear equations can be graphed as lines on the coordinate plane (rectangular coordinate system).
Example:
x/3 = -x + 1 is a linear equation in 1 variable that can be simplified to x =3/4.
On the coordinate plane, the graph of the above linear equation would look like below:
Similarly, a linear equation in 2 variables, 2x+3y=6 can be plotted on the coordinate plane as shown below:
In general, the process to plot a graph manually would be to find the points that lie on the graph/equation and join them.
In the second example, if we substitute x with 3, y would equal 0 and if we substitute x with 0, y would equal 2. Hence, the paired coordinates (3,0) and (0,2) lie on the line which can be seen in the graph too.
A table of paired values of x, y which satisfy the equation is shown below:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 4 | 10/3 | 8/3 | 2 | 4/3 | 2/3 | 0 |
All the points that lie on the line (graph) that can be derived from the above table would hence be (-3,4), (-2,10/3), (-1,8/3), (0,2), (1,4/3), (2,2/3) and (3,0) lie on the line shown in the graph.
A line containing points in which the value of y decreases as the value of x increases is negatively sloped (like the one graphed above) and shows a negative correlation between the 2 variables.
Plotting such a linear graph for another linear equation in 2 variables will help us understand the solution for the pair of linear equations if any.
If the lines graphed intersect, the point of intersection itself is the solution of the pair of linear equations. If the lines so plotted turn out to be parallel to each other, there exists no point of intersection and if the lines so plotted turn out to be coinciding(on over another), there exist infinitely many solutions.
A pair of linear equations which does not have a solution (parallel lines) is shown below:
The equations 2x+3y=6 and 4x+6y=10 are plotted as shown and are hence parallel and do not have a solution (point of intersection).
The equations 2x+3y=6 and 4x+6y=12 are plotted as shown and are hence coinciding and have infinitely many solutions.
In general, the standard format of a linear equation is ax+by+c=0 where x, y are variables and a, b, care constants.
If ax+by+c=0 and px+qy+r=0 are a pair of linear equations in 2 variables, then, they would have a unique solution if a/b is not equal t p/q; no solution if a/b = p/q but not equal to c/r; infinitely many solutions if a/p=b/q=c/r.
You may try:
If 2x-3y=5 and px-6y=r are two linear equations which have infinitely many solutions what is the value of r-p?
