Solving a system of linear equations using a graph
A system of linear equations includes two or more linear equations. The graphs of two lines will intersect at a single point if they are not parallel. Two parallel lines can also intersect if they are coincident, which means they are the same line and they intersect at every point. For two lines that are not parallel, the single point of intersection will satisfy both equations and therefore represent the solution to the system.
To find this point when the equations are given as functions, we can solve for an input value so that
In other words, we can set the formulas for the lines equal to one another, and solve for the input that satisfies the equation.
Finding a point of intersection algebraically
Find the point of intersection of the lines
and
Set
This tells us the lines intersect when the input is
We can then find the output value of the intersection point by evaluating either function at this input.
If we were asked to find the point of intersection of two distinct parallel lines, should something in the solution process alert us to the fact that there are no solutions?
Yes. After setting the two equations equal to one another, the result would be the contradiction “0 = non-zero real number”.
Look at the graph in
[link] and identify the following for the function
y- intercept
x -intercept(s)
slope
Is
parallel or perpendicular to
(or neither)?
Is
an increasing or decreasing function (or neither)?
Write a transformation description for
from the identity toolkit function
Slope -1
Neither parallel nor perpendicular
Decreasing function
Given the identity function, perform a vertical flip (over the
t -axis) and shift up 5 units.
A company sells sports helmets. The company incurs a one-time fixed cost for $250,000. Each helmet costs $120 to produce, and sells for $140.
Find the cost function,
to produce
helmets, in dollars.
Find the revenue function,
from the sales of
helmets, in dollars.
Find the break-even point, the point of intersection of the two graphs
The cost function in the sum of the fixed cost, $125,000, and the variable cost, $120 per helmet.
The revenue function is the total revenue from the sale of
helmets,
The break-even point is the point of intersection of the graph of the cost and revenue functions. To find the
x -coordinate of the coordinate pair of the point of intersection, set the two equations equal, and solve for
To find
evaluate either the revenue or the cost function at 12,500.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits