In this explainer, we will learn how to find the intersection point between two straight lines on a coordinate system and use this concept to find equations of lines.
We begin by recalling what we mean by the intersection of two lines.
Definition: The Point of Intersection of Two Lines
The point of intersection of two distinct, nonparallel lines is the single point where they meet or cross. It is the ordered pair of the values of and where the lines meet on the graph and that satisfies the equations of both lines.
Distinct, parallel lines are lines in a plane that are always the same distance apart. They will have no points of intersection.
In the diagram below, and intersect at point .
How To: Finding the Point of Intersection between Two Lines on the Coordinate Plane
We can find the point of intersection of two lines on the coordinate plane using two methods, a graphical or an algebraic method. For example, if we had the two lines represented by and , then we could graph these as follows.
The specific point of intersection can be found graphically by inspecting the graph to note the point , where the lines meet or cross. However, for more complex equations, an algebraic solution may be preferable. For an algebraic solution, we recognize that a point lying on both lines must satisfy both equations. This is the same as solving a system of two linear equations in two unknowns.
In the first example, we will see how a graphical solution can be used to find the point of intersection between a horizontal and a vertical line.
Example 1: Finding the Point of Intersection of a Horizontal and a Vertical Line
At which point do the lines and intersect?
Answer
The intersection point of two distinct lines is the point where the lines cross. One way to answer this question is to sketch both lines. We begin by drawing the graph of . Any equation that just has an -variable, and no -variable, will be a vertical line.
Next, we consider the graph of . It may be helpful if we rearrange this to make the subject. So, by multiplying both sides of the equation, , by 6, we have
An equation with just a -variable is a horizontal line. As , the horizontal line passes through on the -axis.
To find the intersection point of the two lines, and , we look for the point where they cross or meet. Inspecting the graph, the intersection point occurs at the coordinate
In the next example, we will see how an algebraic method can be used to find the point of intersection between two lines.
Example 2: Finding the Point of Intersection of Two Straight Lines
Determine the point of intersection of the two straight lines represented by the equations and .
Answer
We can recall that the intersection point of two distinct lines is the point where the lines cross. To find this point of intersection, or the point where the lines cross, we can use either an algebraic or a graphical approach.
A point of intersection lies on both lines, so it must satisfy the equations of both lines. Hence, we can find the intersection point by solving these equations as simultaneous equations, finding the values of and , where is the point of intersection.
We can write our equations:
To solve this using a substitution method, we rearrange the second equation, , by adding to both sides of the equation to give
Substituting into the equation (1), and rearranging, we have
We have the solution and . Therefore, the point of intersection can be given as
As an alternative method, or as a check on the algebraic method, we consider the graphs of the two lines.
We have established that the line can be rearranged as . Hence, the line will be a horizontal line passing through 1 on the -axis.
We can sketch the equation by finding two points on the line. For example, we could find the -intercept by substituting and the -intercept by substituting .
Substituting into , and simplifying, we have
Substituting , and simplifying, gives
We have now established that the line passes through the points and . It can be a little difficult to plot a set of coordinates with noninteger values, such as ; so, we may prefer to find another set of coordinates on the line .
Substituting into , and simplifying, we have
This gives us a third coordinate, , on the line . We can plot these three coordinates and sketch this line alongside the graph of as shown.
Inspecting the graph, we can confirm the algebraic solution above, as the intersection point is at
In the previous example, we saw two different methods used, an algebraic and a graphical one. There are advantages to both methods, and often a graphical method is a good way to check on the result of an algebraic one. However, it is worth noting that a graphical method may not be fully accurate, particularly in the cases where the solution is a noninteger result. On the other hand, an algebraic solution will always give an accurate result.
In the next example, we will see how the intersection can be found between two lines, where one is given in vector form.
Example 3: Finding the Vector Form of a Line That Passes through the Intersection of Two Other Lines
Find the vector equation of the straight line that is parallel to the -axis and passes through the point of intersection of the two straight lines and .
Answer
We recall that the intersection of two lines is the point at which they meet or cross. In order to find the vector equation of a line, we need a point on the line and its direction. Since the line is parallel to the -axis, it is a vertical line and has the direction vector . Therefore, we need to find a point on the line. This point will be the point of intersection.
We can write the line with vector equation as one in Cartesian form. For any points and in the coordinate system with a direction vector , for any scalar ,
Hence, taking , we can write the equation in parametric form as
We can then eliminate by rearranging each equation above to make the subject. This gives us
Thus, we can set the left-hand side of each equation as equal, such that
Rearranging, we have
Next, the line can be rearranged as , and we can find the point of intersection of the two lines in Cartesian form using a simultaneous method:
The second equation can be rearranged to make the subject, as follows:
We then substitute this value into equation (3) to give
Now, we can substitute this value of into to give
Thus, the point of intersection of the two lines is at .
We now have the vector direction, , as the unit vector and the position vector of the point of intersection as .
Writing this equation in the form , where as any scalar multiple, we have
In the next example, we will use a given angle to find the direction of a line through the intersection of two other lines.
Example 4: Finding the Equation of a Straight Line That Passes through the Intersection of Two Other Lines
Determine the equation of the line passing through the point of intersection of the two lines whose equations are and while making an angle of with the positive -axis.
Answer
We recall that the intersection of two distinct lines is the point where they cross. We are given the information about the angle that the required line makes with the positive -axis; however, we will also need to find a point on the line in order to write the equation of this line. As the point of intersection of the other lines lies on this line, then this will be an ideal point to choose.
We begin by finding the intersection of the lines and . At the intersection point, the - and -values in each equation will be equal. We can find these values by using a substitution method.
Taking the equation , and rearranging to make the subject, we have
We can then substitute into the equation , and rearrange, to give
To find the value of , we substitute into our rearranged equation, . This gives us
The point of intersection, , can be given as
We need to find the equation of the line passing through this point while making an angle of with the positive -axis. We can sketch a line with an angle of to the -axis. As the angle is positive, the measurement is taken counterclockwise.
To find the slope of the line, we need to calculate the positive angle that the line makes with the positive direction of the -axis. We can use the angles on a line and the corresponding angles resulting from parallel lines and transversals.
We remember that the angles on a straight line sum to . Therefore, the line will make an angle of to the positive -axis, measured in a clockwise direction.
Hence, the line will make an angle of with the positive -axis, measured in a counterclockwise direction.
If the line makes an angle with the positive -axis, then its slope is .
The slope, , is
We can use the pointβslope form of a line to write the equation of this line. In this form, the equation of a line passing through , with slope , is given by
So, the line passing through the point , with slope 1, is
Adding to both sides of this equation gives
Then, multiplying the terms by 29, we have
Writing this equation in the form , we subtract and 91 from both sides of the equation to give
We can now consider how we can write the general equation of the straight line passing through the point of intersection of two given lines.
We recognize that there is an infinite number of straight lines passing through any particular point. Hence, we can define the equation that represents all straight lines passing through an intersection point of two lines as follows.
Definition: The Equation of a Straight Line Passing through the Point of Intersection of Two Given Lines
The equation that represents all straight lines passing through an intersection point of two lines and is where .
If , we have the equation of the second line.
If , we have the equation of the first line.
When and , we have the equation of a straight line passing through the point of intersection, excluding the original lines. Hence, we can write the equation above in the form for any .
In the next example, we will see how we can apply this equation in an algebraic method to find the equation of a line passing through a given point and the intersection point of two lines.
Example 5: Finding the Equation of a Line That Passes through the Intersection of Two Other Lines
What is the equation of the line passing through and the intersection of the lines and ?
Answer
We begin by recalling that the intersection point of two distinct lines is the point where they cross.
We can write the general equation of a line passing through the intersection point of two lines and as for any .
Substituting the values , , and as the values from the line and the , , and values from the line , we have
As the line passes through the point with coordinates , we can substitute and into equation (5) above. This gives us
We can now substitute into equation (5). This gives us
Therefore, the equation of the line passing through and the intersection of the lines and is
As an alternative method, we can find the equation of a line by using two distinct points on the line. Therefore, we use the intersection point and point to find the equation of the line.
We can find the intersection of and by solving the equations simultaneously using the elimination method. Using an elimination method. We can number our equations as
In order to eliminate either the - or -variables, their absolute values need to be the same in both equations. We observe that we can multiply equation (6) by 2 in order to have the same absolute value of in each equation. Thus, we have
We then eliminate by adding the two equations (8) and (9):
Next, rearranging by subtracting 13 from both sides of the equation, and then dividing by 11, gives us
We have now found the -coordinate of the point of intersection, so substituting this into either of the equations (6) or (7) would allow us to find the -coordinate. Substituting into equation (6), and simplifying, we have
This gives us the point of intersection
Now, we need to find the equation of the line passing through the points and .
We recall that a line containing a coordinate , with slope , is given in pointβslope form as
To use this form, we need to calculate the slope, , of a line joining two points and , which is given by
We can designate as and as . Substituting these in, we have
We now have the slope of the line being , along with two points that lie on the line. We only need one of these points to be able to use the pointβslope form of an equation alongside the slope.
Thus, substituting for the point and the slope into the equation , we have
Distributing across the parentheses on the right-hand side, and then adding 3 to both sides of the equation, gives us
This is a valid equation for the line. However, we can also write this in the general form of the equation of a line, , where .
We multiply all the terms by 2 and then subtract from both sides of the equation, which gives
This confirms the answer we found using the first method; the equation of the line is
We can now summarize the key points.
Key Points
- The point of intersection between two distinct lines is the point where they meet or cross. It is the ordered pair of the values of and where the lines meet on the graph and that satisfies the equations of both lines.
- Distinct, parallel lines are lines in a plane that are always the same distance apart. They will have no points of intersection.
- We can find the intersection of two lines either graphically or algebraically. Algebraic solutions will always give an accurate result.
- To find the point of intersection between two nonparallel lines, algebraically, we solve the system of two equations. The solution values of and form the intersection point .
- The equation that represents all straight lines passing through an intersection point of two lines and is where . When and , then for any ; the equation of a straight line passing through the point of intersection can be written as
