Find the possible slopes of a line that passes through (4,3) so that the portion of the line in

Find the possible slopes of a line that passes through (4,3)...

Key Concepts

Algebraic Constraint from Geometric Conditions

This key concept involves setting up algebraic equations based on geometric properties or conditions. For instance, when a certain area is given for a triangle formed by intercepts, it leads to an equation that relates the slope and intercepts of a line, enabling the determination of unknown parameters such as the slope.

Area of a Triangle Formed by Axes Intercepts

When a line crosses the coordinate axes, the triangle formed with these intercepts and the origin has an area that can be calculated by the formula (1/2)*|x-intercept|*|y-intercept|. This concept links geometric area with algebraic expressions and is central in problems imposing area constraints.

Line Equation in Slope-Intercept Form

This concept describes how a line can be represented algebraically in the form y = mx + b, where m is the slope and b is the y-intercept. It is essential for analyzing how changes in these parameters affect the position and orientation of the line in the coordinate plane.

Intercepts with the Coordinate Axes

Intercepts are the points where a line crosses the x-axis and y-axis. Understanding intercepts is crucial because they define the endpoints of the segment of the line that lies within a specific quadrant, and are used to form geometric figures such as triangles with the axes.

Transcript

00:02 All right.

00:04 Our question here is to find the equation of a line that passes that has a slope of negative 5 and it goes through point 3 .3 .6.

00:29 Okay.

00:30 So let's do it.

00:31 That's first.

00:32 And then the question is to find the area of the triangle that is formed by this line.

00:39 And the xx xx y x so first things first let's find the equation of the line let it be out of the form y equals mx b m is given so that's negative 5 x plus b let's use 3 6 there's no choice you can only use 3 6 and plug them into y and x and get b which means so 3 6 using b 6 okay equals next negative 5x plus b 5 is 6 equals 5 i 5 plus b which means b equals 6 plus 15 equals 21 b is 21 okay so our equation is finally y equals minus 5 times 6 plus 21 okay next to find the area of we need to find i like that section b navy we need to find the area of the triangle find the area with x and y so obviously obviously we need a need to what will help us negative 5x plus 21 and i always always know the best way to plot the line is to use to find the x and y intercept in other words to make the to make x equal 0 and find y and then make y equal 0 and find x and then you have no then you have two points this is rather line connecting those two points so okay so first point i'll make x 0 which means y is 21 so so x equal 0 let's say y is 21 that's 0 21 and next i'll make y equal to 0 0 0 0 0 0000 and next i'll make y equal to so y equal to 0 means negative 5x plus 21 equal 0 or 5x equals 21 and x equals 21 over 5.

03:31 That's a smaller value.

03:33 So it's just if for the same scale, it's about one -fifth of it.

03:39 Maybe around here.

03:44 That would be 21 over 5...

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