Category Archives: Algebra

Using the Distance Formula

March 25, 2017 admin

The Distance Formula

Suppose A is ( x_1, y_1) and B is ( x_2, y_2)

The distance between points A and B is given by the following formula.

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

Example: Find the y-coordinate of the points that are 5 units away from the point (3,3) that have an x-coordinate of -1.

$5=sqrt {(-1-3)^2+ (y-3)^2}$

$5=sqrt {(-4)^2+ (y-3)^2}$

$5=sqrt {16+ (y-3)^2}$

$5^2=(sqrt {16+ (y-3)^2})^2$

25=16+ (y-3)^2

25-16=16-16+ (y-3)^2

9= (y-3)^2

$sqrt{9}= sqrt{(y-3)^2}$

pm 3= y-3

3 pm 3= y

3 + 3= y or 3 - 3= y

6= y or 0= y

2.1 Graphing, 3.6 Mathematical Models: Building Functions, Distance

Finding the Distance

March 25, 2017 admin

The Distance Formula

Suppose A is ( x_1, y_1) and B is ( x_2, y_2)

The distance between points A and B is given by the following formula.

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

Example: Find the distance between points A and B.

Point A is ( 4, -2) and point B is ( -8, 3)

$d=sqrt {(-8-4)^2+ (3-(-2))^2}$

$d=sqrt {(-12)^2+ (3+2)^2}$

$d=sqrt {144+ (5)^2}$

d=sqrt {144+ 25}

d=sqrt {169}

d=13

2.1 Graphing, Midpoint

Finding the Midpoint

March 25, 2017 admin

The Midpoint Formula

Suppose A is ( x_1, y_1) and B is ( x_2, y_2)

The midpoint of the line segment joining points A and B is given by the following formula.

$( {x_1+x_2}/2, {y_1+y_2}/2)$

Example: Find the midpoint of a line segment joining points A and B.

Point A is ( 4, -2) and point B is ( -8, 3)

The x coordinate of the midpoint can be found by averaging the x-values of point A and B. The y-coordinate of the midpoint can be found by averaging the y-values of point A and B.

( {4+(-8)}/2, {-2+3}/2)

( {-4}/2, {1}/2)

( -2, {1}/2)

2.2 Lines, 2.3 Lines, Linear Equations

Forms of Linear Equations

March 19, 2017 admin

Slope Intercept Form

y=mx+b

m is the slope of the line and (0,b) is the y-intercept

Point Slope Form

y-y_1=m(x-x_1)

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

Standard Form of a Line

Ax+By=C

2.2 Lines, 2.3 Lines, 4.1 Linear Functions and Their Properties, Graphing with Slope Intercept Form

Graphing Linear Equations with Slope Intercept Form

March 19, 2017 admin

Example: Graph the linear equation using the slope and the y-intercept. (the slope is positive and fractional)

Example: Graph the linear equation using the slope and the y-intercept. (positive and negative fractional slopes)

Example: Graph the linear equation using the slope and the y-intercept. (equation written in standard form)

2.2 Lines, 2.3 Lines, Graphing By Plotting Points

Graphing Horizontal and Vertical Lines

Video March 19, 2017 admin

2.2 Lines, 2.4 Parallel and Perpendicular Lines, Slope

Perpendicular Lines and Parallel Lines

March 19, 2017 admin

Example: What are parallel and perpendicular lines?

Example: How are the slopes of parallel and perpendicular lines related? (only watch until 1 min 20 seconds)

Example: Are the lines parallel, perpendicular or neither?

Example: Are the lines perpendicular to each other?

2.2 Lines, 2.4 Parallel and Perpendicular Lines, Finding the Equation of a Line

Finding the equation of a line perpendicular to another line

March 19, 2017 admin

Example: Write the equation of a line in point-slope form passing through the point (-3,9) and perpendicular to the line whose equation is y={6/5}x+9/5 .

Solution:

Use the point-slope formula of the line to start building the line. m represents the slope of the line and (x_1,y_1) is a point on the line.

Point-slope formula: y-y_1 = m(x-x_1)

Although the slope of the line is not given, the slope can be deducted from the line being perpendicular to y={6/5}x+9/5 .

Perpendicular lines have negative reciprocal slopes. Since the slope of the given line is 6/5 , the slope of the perpendicular line -5/6 .

m=-{5/6} and (-3,9)

Substitute the values into the point-slope formula.

y-9 = {-5/6}(x-(-3))

The point-slope form of the line is as follows.

y-9 = {-5/6}(x+3)

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope intercept form)

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope standard form)

Example: Find the equation of a line perpendicular to the x-axis.

Example: Find the equation of a line perpendicular to the x-axis and perpendicular to the y-axis.

2.2 Lines, 2.4 Parallel and Perpendicular Lines, Finding the Equation of a Line

Finding the Equation of a Line parallel to another line

March 19, 2017 admin

Example: Find the equation of the line parallel to another line and passing through a specific point. (parallel equation in slope intercept form)

Example: Find the equation of the line parallel to another line and passing though a specific point. (parallel line in standard form)

Example: Find the equation of the line parallel to the x-axis or y-axis and passing through a specific point.

Example: What is an equation parallel to the y-axis?

Example: What is an equation parallel to the x-axis?

2.2 Lines, 2.3 Lines, Finding the Equation of a Line

Finding the Equation of a line given a fractional slope and a point

March 18, 2017 admin

Example: Find the equation of a line in slope intercept form given the slope of the line is -{2/3} and the line passes through the point (-4,7)

Solution:

Use the point-slope formula of the line to start building the line. m represents the slope of the line and (x_1,y_1) is a point on the line.

Point-slope formula: y-y_1 = m(x-x_1)

m=-{2/3} and (-4,7)

Substitute the values into the formula.

y-7 = -{2/3}(x-(-4))

Since the instructions ask to write the equation in slope intercept form (y=mx+b) we will simplify and write the equation with y by itself on one side. I will also use the clearing fractions method to avoid having to add fractions.