Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of 15 feet. H
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Answer:
Step-by-step explanation:
The given expression is
We want to find the value of h(-9).
We substitute r=-9 to obtain:
We multiply out to obtain:
We simplify the denominator to obtain:
This is the same as:
Explanation:
The solution set for a system of equation is the set of points where the graphs of the equations intersect.
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<h3>general case</h3>A system will have <em>one solution</em> if there is a <em>single point of intersection</em> of the graphs of the equations.
A system will have <em>no solutions</em> if the graphs have <em>no points of intersection</em>.
A system will have an <em>infinite</em> number of <em>solutions</em> if the graphs <em>intersect at an infinite number of points</em>.
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<h3>linear equations</h3>When the equations are linear equations, their graphs are straight lines. If the lines have different slopes, they must intersect at exactly one point: there will be one solution.
If the lines have the same slope, there are two possibilities:
- the lines are parallel -- no solutions
- the lines are coincident -- infinite solutions
The attached graph illustrates these cases.
- the red and blue lines are the graphs of a system of equations with one solution. Those lines have different slopes
- the blue and green lines are the graphs of a system of equations with no solution. Those lines are parallel.
- The red and (dotted) purple lines are the graphs of a system of equations with infinite solutions. Those lines are coincident.
By solving a quadratic equation, we will see that the width of the walkway is 3ft.
<h3>How to get the width of the walkway?</h3>Remember that for a rectangle of length L and width W, the area is:
A = W*L
Now, we know that for our garden we have:
W = 8ft
L = 12ft
If we add a walkway of width x around the garden, then the new measures are:
W' = 8ft + 2x
L' = 12ft + 2x
So the area is:
A' = ( 8ft + 2x)*( 12ft + 2x)
And we know that it is equal to 192 ft^2, then:
192 ft^2 = ( 8ft + 2x)*( 12ft + 2x)
Now we can solve this for x.
192ft^2 = 96ft^2 + 4x^2 + 20ft*x
0 = 96ft^2 - 192ft^2 + 20ft*x + 4x^2
0 = -96ft^2 + 20ft*x + 4x^2
This is a quadratic equation, and the solutions are given by the Bhaskara's formula:
We only take the positive solution, so we get:
x = (-20ft + 44ft)/8 = 3ft
So the width of the walkway is 3ft.
If you want to learn more about quadratic equations, you can read: