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.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?