Math Lesson 14.4.5 - Intersecting Lines

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Welcome to our Math lesson on Intersecting Lines, this is the fifth lesson of our suite of math lessons covering the topic of Parallel, Perpendicular and Intersecting Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Intersecting Lines

Intersecting lines are those lines that intercept each other at a single point but not at right angles. Therefore, in this case, we cannot find the gradient (and the equation) of the other line if one of the lines is known unless we don't have at least two known points of the second line given, where one of them may be the intercept of the two lines.

Example 5

The line L1 with equation y = 3x - 2 is intercepted at point A(-7, 6) by another line L2, which passes through point B(-1, 4). Find the equation of the line L2.

Solution 5

First, let's find the gradient of L2 since we have two points given. Thus, from the equation of gradient

where x₁ and y₁ are the coordinates of the leftmost point (here, of point A), we obtain the gradient m₂ of the second line

Now, we can substitute the coordinates of any known point (for example, of the intercept A) in the equation of L2

to obtain

In this way, the equation of L2 becomes

Remark! You can multiply both sides of the last equation by 3 and send all terms on the left side. In this way, you obtain the equation of L2 in the form ax + by + c = 0. The same thing can be done with the line L1 as well. In this way, we get rid of fractions in the equations of the system. This is a huge advantage, as the system is solved easier. However, the disadvantage of this method besides the extra time needed to do this transformation consists of the fact that the variable y is not isolated anymore. This may cause problems in identifying the points required.

From all we have discussed in this tutorial, we draw four important conclusions about the relationship between linear graphs:

1. Parallel lines have the same gradient and no intercepts. Hence, the system of linear equations formed by them has no solution.
2. Perpendicular lines represent a special case of intersecting lines, where the product of the corresponding gradients is -1.
3. If two lines have different gradients m₁ and m₂, and their product is different from -1, they are simply intersecting.
4. If the coefficients m and constants n of two lines are the same, they overlap.

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2. Linear Graphs Math tutorial: Parallel, Perpendicular and Intersecting Graphs. Read the Parallel, Perpendicular and Intersecting Graphs math tutorial and build your math knowledge of Linear Graphs
3. Linear Graphs Revision Notes: Parallel, Perpendicular and Intersecting Graphs. Print the notes so you can revise the key points covered in the math tutorial for Parallel, Perpendicular and Intersecting Graphs
4. Linear Graphs Practice Questions: Parallel, Perpendicular and Intersecting Graphs. Test and improve your knowledge of Parallel, Perpendicular and Intersecting Graphs with example questins and answers
5. Check your calculations for Linear Graphs questions with our excellent Linear Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Linear Graphs Calculators by iCalculator™ below.
6. Continuing learning linear graphs - read our next math tutorial: Line Segments

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