Consider the following pairs of lines. Determine whether the lines are parallel or intersecting.

step 1 So for part A, we have parametric equations. X is equal to 1 plus S and Y is equal to 2S. And th

Consider the following pairs of lines. Determine whether the...

Key Concepts

Parametric Representation of Lines

This concept refers to expressing the coordinates of points on a line using one or more parameters. In the parametric form, each coordinate is defined as a function of a parameter, which allows for an elegant description of the line and facilitates calculations for intersections or other relationships between lines.

Direction Vectors

The direction vector of a line indicates the direction in which the line extends. In the context of parametric equations, these vectors are extracted from the coefficients of the parameter. They play a crucial role in determining the relationship between two lines, such as checking for parallelism or finding angles between them.

Parallel Lines

Two lines are parallel if their direction vectors are scalar multiples of each other. This means that the lines share the same slope or orientation, implying that they either never meet or are coincident. Checking the proportionality of the direction vector components offers a straightforward method to test for parallelism.

Intersection of Lines

The intersection of lines occurs when there is a common point that satisfies the parametric equations of both lines. To find this point, the individual equations for x and y are equated and solved as a system of equations. If a solution exists, it represents the point where the two lines meet.

Solving Systems of Equations

This concept involves finding values for the parameters in the line equations that make both equations true simultaneously. By setting the corresponding x and y expressions equal to each other, one forms a system of linear equations, which can then be solved using methods such as substitution or elimination to determine whether and where the lines intersect.

Transcript

00:01 So for part a, we have parametric equations.

00:05 X is equal to 1 plus s and y is equal to 2s.

00:11 And the other equations we have are x is equal to 1 plus 2t and y is equal to 3t.

00:19 So what we want to do here is we want to figure out whether these two lines are parallel or not.

00:25 And if they are not parallel, then where do they intersect? so from here, what we can say is s is equal to x minus 1, which we can substitute into y, so that would give me y is equal to two times x minus 1, which is y is equal to 2x minus 2.

00:45 In the same way here, we can isolate t, and t would be equal to x minus 1 over 2, and substituting that into y would give me 3 times x minus 1 over 2.

00:58 2 and that gives me x is equal to 3 or y is equal to 3x over 2 minus 3 over 2 so since the slopes are not the same therefore these two lines are not parallel now that we've established that they are not parallel we would want to find their point of intersection so we can just equate 2x minus 2 with 3x over 2 minus 3 over 2 and that would simplify to x over 2 minus 2 minus 2 is equal to negative 3 over 2 and that gives you x over 2 is equal to 1 over 2 and that means that x is equal to 1.

01:46 So the point where they intersect would be the point where x is equal to 1 and then we can say y is equal to 2 times 1 minus 2 because earlier one of the equations we had was y is equal to 2x minus 2 so that gives you y is equal to 0 so we have our point of intersection and the point of intersection is the point 1 0 so let's take a look at part b part b again gives you two sets of equations and you have to figure out whether they're parallel or intersecting.

02:27 So we have x is equal to, we have x is equal to 2 plus 5s and y is equal to 1 plus s.

02:38 Similarly we have x is equal to 4 plus 10 t and y is equal to 3 plus 2 t.

02:49 So same way.

02:51 We're going to isolate s here so s is equal to x minus 2 over 5 and substitute that into y so y is equal to 1 plus x minus 2 over 5 y is equal to 5 plus x minus 2 over 5 and that gives you y is equal to x plus 3 over 5 or y is equal to x over 5 plus 3 over 5 in the same way we come here from here we can say t is equal to x minus 4 over 10 substitute that into y so y is equal to 3 plus 2 times x minus 4 over 10 so the 2 and the 10, they simplify to give you 5.

03:51 And now we can say y is equal to 15 plus x minus 4 over 5, which simplifies to y is equal to x plus 11 over 5.

04:07 And then y is equal to x over 5 plus 11 over 5.

04:12 Since both these lines have the same slopes, therefore these lines are.

04:16 Parallel and if they are parallel then they do not intersect so now let's go to part c in part c again we're given these two equations x is equal to 1 plus 3s and y is equal to 4 plus 2s and then x is equal to 4 minus 3 t y is equal to 6 plus 4 t so we are going to isolate s here so so s is equal to x minus 1 over 3.

04:52 So y is equal to 4 plus 2 times x minus 1 over 3, which simplifies to y is equal to 8 plus 2 times x minus 1 over 3.

05:11 Let's simplify that...

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