Contrapositive Statements

Introduction:

The intersection of lines is when two or more lines cross in a plane. Each intersecting line converges on a single point called the point of intersection, which can be found on all intersecting lines. Only if the lines are not parallel will they intersect. A road cross, a folding chair, a signboard, and a pair of scissors are all examples of intersecting lines in the real world.

a1x + b1y + c1=0 and a2x+b2y+c2=0 are the equations for these two lines, respectively. Point O, the point of intersection, is where lines P and Q meet. The point of intersection is shown in the image below: 

Point of Convergence or Intersection in the intersection of lines study material: 

Have you ever encountered a traffic sign like this while driving?

• Seeing this sign indicates that you’re at an intersection where two roads meet. As you can see, the intersection traffic sign has two intersecting lines that meet in the middle. This is where they meet. In mathematics, the point where two lines or curves meet is a point of intersection.
• When two curves meet, it’s essential to observe that they have the same value at the intersection. This can be useful in a wide range of circumstances. Suppose we’re working with two equations: one describing the revenue of a business and the other representing its expenses. The breakeven point is found at the junction of the two curves that correspond to these two equations.

The properties of Intersection of Lines:

• Two or more intersecting lines always meet at the same place.
• Any angle can be used to span the intersection of lines. This angle is always more significant than 0° and less than 180°; this angle is generated.
• Two intersecting lines form a pair of vertical angles. The vertex of both the vertical and horizontal angles is the same (the point at which the lines intersect).
• Display style x-coordinates and y-coordinates describe only one point on a two-dimensional graph where straight lines meet. If the x-coordinates and y-coordinates of the display style travel through that point, then both equations must be satisfied.

How do I determine the point for the intersection of lines?

To find a point for the intersection of lines, we can graph the curves and find their points of intersection on the same graph.

To find an algebraic point of intersection, perform these steps:

• Set up a slope-intercept form for the two linear equations. In other words, prepare them as follows: It is easy to see that y= mx+b. 
• Make y the same in both equations.
• Find the value of x. This will be the junction point’s x-coordinate.
• To solve for y, substitute this x-coordinate into one of the original equations for the lines. They-coordinate of the junction point will be this.
• They-coordinate should be the same if you substitute the x-coordinate into the other equation.
• You now know the x and y coordinates for the point of intersection.

A few solved examples on the intersection of lines:

1. How to calculate the intersection of lines of two lines x + 2y + 1 = 0 and 2x + 3y + 5 = 0?

Solution:

Here,

• a1 = 1, b1 = 2, c1 = 1,
• a2 = 2, b2 = 3, c2 = 5

So, we can calculate the points of intersection using this formula:

x = b1c2−b2c1/a1b2−a2b1

 y = a2c1−a1c2/a1b2−a2b1 

(x,y) = (2×5−3×1/1×3−2×2 ,2×1−1×5/1×3−2×2)

(x,y) = (10−3/ 3−4, 2−5/3−4)

(x,y) = (-7, 3)

2. How to find the point of intersection of lines for the two lines mentioned below? 

• x – 2y + 3 = 0
• 3x – 4y + 5= 0.

Solution:

A Cramer’s rule is used to find the intersection of the lines:

x/ (-10 – (-12)) = -y/(5-9) = 1/(-4 – (-6))

⇒ x/2 = y/4 = 1/2

⇒ x = 1, y = 2

The slope for the two intersections of lines is now: m1=1/2 and m2=3/4.

 When the acute angle for these two intersection of lines is θ, then we get:

tan θ=∣m2−m1/1+m1m2∣=∣3/4−1/2/1+3/8∣ =2/11

 θ = tan−1 (2/11) ≈ 10.3∘

The two intersections of lines meet is= (1, 2). 

When two lines meet, the angle at which they meet θ= tan-1 (2/11)

Other Points of Intersections:

Two-line intersections are not the only type of intersections. We can find the point of intersection between any two curves. More than one junction may be located if we don’t limit ourselves to lines. If you combine a function with another function, you may get infinitely many intersections.

Conclusion:

Here, we learned about the intersection of lines and concepts with the principles and a few solved examples. Find the point where y = ax + b intersects c and d – set ax + b to equal cx + d as the very first step to solving the equation. Then, find x by solving for this equation. This value will give the intersection point’s x coordinate. Fill in the x coordinate in the expression of either of the two lines to get the y coordinate of the intersection. In this case, both points will have the same y coordinate because it is an intersection.

Also, we have come to know about a few other types of intersections. There may be multiple intersections in these situations. To find x, simply set both expressions to equal values and use the same procedure as before. By substituting x into an expression, you may then calculate y.

Some of these examples will be covered in your study material notes on the intersection of lines, and these are just briefly explained here.