the coordinates of point of intersection of the points A X + B Y is equal to 7 and b x + A Y is equal to 5 is (3, 1 ) find the value of A and B How will you determine by comparing A1 by A2 and B1 by B2 that whether the given two lines intersect each other or not write in your own words.
Answers
Answer:
Step-by-step explanation:
An important topic of high school algebra is "the equation of a line." This means an equation in x and y whose solution set is a line in the (x,y) plane.
The most popular form in algebra is the "slope-intercept" form
y = mx + b.
This in effect uses x as a parameter and writes y as a function of x: y = f(x) = mx+b. When x = 0, y = b and the point (0,b) is the intersection of the line with the y-axis.
Thinking of a line as a geometrical object and not the graph of a function, it makes sense to treat x and y more evenhandedly. The general equation for a line (normal form) is
ax + by = c,
with the stipulation that at least one of a or b is nonzero. This can easily be converted to slope-intercept form by solving for y:
y = (-a/b) + c/b,
except for the special case b = 0, when the line is parallel to the y-axis.
If the coefficients on the normal form are multiplied by a nonzero constant, the set of solutions is exactly the same, so, for example, all these equations have the same line as solution.
2x + 3 y = 4
4x + 6y = 8
-x - (3/2) y = -2
(1/2)x + (3/4)y = 1
In general, if k is a nonzero constant, then these are equations for the same line, since they have the same solutions.
ax + by = c
(ka)x + (kb)y = kc.
A popular choice for k, in the case when c is not zero, is k = (1/c). Then the equation becomes
(a/c)x + (b/c)y = 1.
Another useful form of the equation is to divide by |(a,b)|, the square root of a2 + b2. This choice will be explained in the Normal Vector section.
Exercise: If O is on the line, show that the equation becomes ax + by = 0, or y = mx.
Exercise: Find the intersections of this line with the coordinate axes.
Exercise: What is the equation of a line through (0,0) and a point (h,k)?
Finding the equation of a line through 2 points in the plane
For any two points P and Q, there is exactly one line PQ through the points. If the coordinates of P and Q are known, then the coefficients a, b, c of an equation for the line can be found by solving a system of linear equations.
Example: For P = (1, 2), Q = (-2, 5), find the equation ax + by = c of line PQ.
Since P is on the line, its coordinates satisfy the equation: a1 + b2 = c, or a + 2b = c.
Since Q is on the line, its coordinates satisfy the equation: a(-2) + b5 = c, or -2 a + 5b = c.
Multiply the first equation by 2 and add to eliminate a from the equation: 4b + 5b = 9b = 2c + c = 3c, so b = (1/3)c. Then substituting into the first equation, a = c - 2b = c - (2/3)c = (1/3)c.
This gives the equation [(1/3)c]x + [(1/3)c}y = c. Why is the c not solved for? Remember that there are an infinite number of equations for the line, each of which is multiple of the other. We can factor out c (or set c = 1 for the same result) and get (1/3)x + (1/3)y =1 as one choice of equation for the line. Another choice might be c = 3: x+y = 3, which has cleared the denominators.
This method always works for any distinct P and Q. There is of course a formula for a, b, c also. This can be found expressed by determinants, or the cross product.