P.o.D. – Find the x- and y-intercept(s) for each.
1.) y=5x-6
2.)
3.)
4.)
5.)
6.)
1.) x=6/5, y=-6
2.) x=-4, y=2
3.) x= ½ , y= None
4.) x=-10, y=-10
5.) x=0 and 2, y=0
6.) x=6,
1.3 – Linear Equations in Two Variables
Learning Target(s): I can use slope to graph linear equations; find slopes of lines; write linear equations in two variables; use slope to identify parallel and perpendicular lines; use slope to model real-life problems
The Slope-Intercept Form of a Line:
Y=mx+b, where m is the slope and b is the y-intercept.
EX: Sketch the graph of -4x-y+5=0.
Begin by putting the equation in slope-intercept form.
-4x+5=y y=-4x+5, so m=-4 is the slope and b=5 or the ordered pair (0,5) is the y-intercept.
Now graph the line.
We could also have graphed the previous line using properties of a Standard Form line.
Ax+By=C
x-int = C/Ay-int= C/Bslope= -A/B
Finding Slope Given Two Points:
, where . If the difference of the x-coordinates is zero, we have undefined slope (a vertical line).
EX: Find the slope of the line passing through (-5,-6) and (2,8).
*Let’s add a part to our distance program that will calculate slope.
EX: Find the slope intercept form of the equation of the line that has a slope of 2 and passes through the point (3,-7). Use this information to find three additional points on the line.
Y=mx+b y=2x+b
-7=2(3)+b
-7=6+b
-13=b
y=2x-13
Three Additional Points:
Y=2(4)-13=-5 (4,-5)
Y=2(5)-13=-3 (5,-3)
Y=2(6)-13=-1 (6,-1)
Point-Slope Form of a Line:
EX: Write the point-slope form a line between (1,-9) and (2, 9).
First find the slope.
Now pick either point and substitute into the equation.
or
Parallel and Perpendicular Lines:
- Two lines are parallel if and only if (IFF) their slopes are equal.
- Two lines are perpendicular IFF their slopes are opposite reciprocals.
EX: Find the slope-intercept forms of the equations of the lines that pass through the point (-4,1) and are (a) parallel to and (b) perpendicular to the line 5x-3y=8.
In Standard Form, slope can be calculated from . Therefore, the slope of the present line is .
a.)
b.)
Terminology: Slope can also be thought of as “rate of change.”
EX: When driving down a mountainside, you notice warning signs indicating that the road has a grade of 12%. This means that the slope of the road is . Approximate the amount of horizontal change in your position if you note from elevation markers that you descended 2000 feet vertically.
EX: The following are the slopes of lines representing annual sales y in terms of time x in years. Use the slope to interpret any change in annual sales for a 1-year increase in time.
a.)The line has a slope of m=120.
Sales are increasing by 120 units per year.
b.)The line has a slope of m=0
There is no change in sales.
c.)The line has a slope of m= -35.
Sales are decreasing by 35 units per year.
EX: A person purchases a car for $25,290. After 11 years, the car will be replaced. Its value at that time is expected to be $1,200. Write a linear equation giving the value V of the car during the 11 years it will be used.
Consider the points (0, 25290) and (11,1200).
Write the equation of the line through these points.
The point (0,25290) is also the y-intercept, so the equation is y= -2190x +25290
*Let’s write another part for our distance program that will write the equation of a line in slope-intercept form if given two points.
EX: A business purchases a piece of equipment for $34,200. After 15 years, the equipment will have to be replaced. Its value at that time is expected to be $1500.
- Write a linear equation giving the value V of the equipment during the 15 years in which it will be used.
(0,34200), (15,1500)
(Use your calculator program to save time)
Y= -2180t + 34200
- Use the equation to estimate the value of the equipment after 7 years.
Y=-2180(7)+34200=$18940
Upon completion of this lesson, you should be able to:
- Graph a line
- Find the slope of a line
- Write the equation of lines
- Determine if lines are parallel or perpendicular
- Apply slope to real-world problems
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HWPg.343-78 3rds, 98, 109, 112, 129-138