Introduction

This article will guide you through the process of finding the equation of a straight line given a point and a slope using the Point Slope Formula as well as how to identify points that are 5 units away from a given point using the Distance Formula. These concepts are fundamental in understanding linear equations and their geometric representations.

Finding the Equation of a Straight Line

To find the equation of a straight line that passes through a given point and has a specified slope, we can use the Point Slope Formula:

( y - y_1 m(x - x_1) )

where ( (x_1, y_1) ) is the point on the line and ( m ) is the slope.

Example: Finding the Equation

Given a point ( (3, 2) ) and a slope of ( frac{3}{4} ), we can substitute these values into the Point Slope Formula:

( y - 2 frac{3}{4} (x - 3) )

Next, we simplify the equation:

Distribute the slope: [ y - 2 frac{3}{4}x - frac{9}{4} ] Add ( 2 ) to both sides: [ y frac{3}{4}x - frac{9}{4} frac{8}{4} ] Combine the constants: [ y frac{3}{4}x - frac{1}{4} ]

Thus, the equation of the line is:

( y frac{3}{4}x - frac{1}{4} )

Finding Points 5 Units Away from a Given Point

To find the coordinates of points that are 5 units away from a given point, we use the Distance Formula:

( d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} )

Where ( d ) is the distance, and ( (x_1, y_1) ) and ( (x_2, y_2) ) are the coordinates of the points. If we set ( d 5 ) and ( (x_1, y_1) (3, 2) ), we get:

( 5 sqrt{(x - 3)^2 (y - 2)^2} )

Squaring both sides: [ 25 (x - 3)^2 (y - 2)^2 ]

This equation represents a circle centered at ( (3, 2) ) with a radius of 5 units. To find specific points on this circle, we can solve for ( y ) given certain ( x ) values:

Case 1: ( x 3 ) [ (3 - 3)^2 (y - 2)^2 25 ] [ (y - 2)^2 25 ] Take the square root of both sides: [ y - 2 5 quad text{or} quad y - 2 -5 ] Thus: [ y 7 quad text{or} quad y -3 ] So we have the points ( (3, 7) ) and ( (3, -3) ). Case 2: ( y 2 ) [ (x - 3)^2 (2 - 2)^2 25 ] [ (x - 3)^2 25 ] Take the square root of both sides: [ x - 3 5 quad text{or} quad x - 3 -5 ] Thus: [ x 8 quad text{or} quad x -2 ] So we have the points ( (8, 2) ) and ( (-2, 2) ).

Summary

The equation of the line that passes through ( (3, 2) ) with a slope of ( frac{3}{4} ) is:

( y frac{3}{4}x - frac{1}{4} )

The points that are 5 units away from ( (3, 2) ) are:

( (3, 7) ) ( (3, -3) ) ( (8, 2) ) ( (-2, 2) )