Question

Write the equations of two lines that intersect at (1,-2).

Explain your process (in paragraph form).

Answer 1

Two lines that intersect at the point (1,-2) could have equations y = x - 1 and y = -2x + 4, which are derived from the point-slope formula using slopes of 1 and -2, respectively.

Step-by-step explanation:

To write the equations of two lines that intersect at the point (1,-2), we need to consider that the point of intersection must satisfy both equations. The simplest way to do this is to create two different linear equations with different slopes that go through the given point. We can choose arbitrary slopes for the two lines, as long as they are not the same, to ensure the lines are not parallel and indeed intersect.

For instance, we can choose the slope of the first line to be 1. Using the point-slope form, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point, we get y - (-2) = 1(x - 1), which simplifies to y = x - 1. Similarly, for the second line with a different slope, say -2, we get y - (-2) = -2(x - 1), which simplifies to y = -2x + 4. Both of these linear equations intersect at (1,-2).

Answer 2

1. y = 2x

2. y = -3x - 5

Step-by-step explanation:

To write the equations of two lines that intersect at the point (1, -2), we need to consider that each line can be represented in the slope-intercept form: y = mx + b, where m is the slope of the line, and b is the y-intercept. Since the lines intersect at the point (1, -2), this means that the coordinates of that point must satisfy the equations of both lines.

Let's assume the equations of the two lines are y = m1x + b1 and y = m2x + b2. Since both equations must pass through the point (1, -2), we can substitute the coordinates of this point into each equation:

For the first line:

-2 = m1 * 1 + b1

For the second line:

-2 = m2 * 1 + b2

Now we have a system of two equations with two unknowns (m1, b1 for the first line, and m2, b2 for the second line). To find the equations of the lines, we need additional information about the slopes or y-intercepts, or both.

For example, let's assume the slope of the first line is 2, and the y-intercept is 0:

m1 = 2, b1 = 0

Then, the equation of the first line becomes y = 2x.

For the second line, let's assume the slope is -3 and the y-intercept is -5:

m2 = -3, b2 = -5

Then, the equation of the second line becomes y = -3x - 5.

So, the equations of the two lines that intersect at (1, -2) are:

1. y = 2x

2. y = -3x - 5

This explanation outlines the process of determining the equations of two lines that intersect at a specific point by using the slope-intercept form and satisfying the coordinates of the intersection point in each equation. The actual values of slopes and intercepts can vary based on the context of the problem.