Form 1267C Math Solution to #49


for what value of a would the following system of equations have an infinite number of solutions?
2x - y = 8
6x - 3y = 41
A: 2
B: 6
C:8
D:24
E:32
2x - y = 8
6x - 3y = 4a.
Multiply the top equation by 3, to give 6x - 3y = 24. This is almost identical with the bottom equation, except for constants. Forthe system to have an infinite number of solution, the constants must be equal. Hence, 4a = 24, and thus a = 6.

Arcs

Arc
From Latin: arcus "a bow, arch,"
Definition: A portion of the circumference of a circle
Try this Drag one of the orange dots that define the endpoints of the blue arc. The arc will change accordingly.
An arc is a portion of the circumference of a circle. In the figure above, the arc is the blue part of the circle. Strictly speaking, an arc could be a portion of some other curved shape, such as an ellipse, but it almost always refers to a circle. To avoid all possible mistake, it is sometimes called a circular arc.
A straight line is drawn between the end points of the arc would be a chord of the circle.
If the arc length is exactly half the circle, this called a semicircular arc. See Semicircle definition.

Naming and identification

Arcs are named by their endpoints. The blue arc above would be called "arc AB". or "arc BA", the order of the endpoints does not matter. As a shorthand this can be written as the letters AB with a curving line above them
Example:      which is read "arc AB".
Notice that this naming can be ambiguous. For example it may mean themajor arc AB, where you go the long way around the bottom of the circle. Unless stated otherwise, it always means the minor arc - the shortest of the two.
If you want to indicate the major arc, add an extra point and use three letters in the name. For example in the diagram on the right the major arc is indicated bywhich is the long arc from A to B going around the bottom via K.
There are two measures of an arc
  1. The length of the arc
  2. The angle of the arc

Arc length

The length of an arc is the distance along the curved line forming the arc. It would be measured in distance units, such as meters. To indicate this measure, the arc is preceded by the lower case letter L (for 'length'). For eaxmaplewould be read as "the length of the arc AB is 6 inches".    See Arc Length page for more.

Angle measure

The angle measure is the angle formed by the arc at the center of the circle. It is indicated by the small letter M in front of the name. For exampleis read as "the arc AB has a measure of 35 degrees".     See Arc Angle Measure for more.

Attributes

Arc LengthThe length of the curved arc line. See Arc Length page for more.
RadiusRadius of an arcThe radius of the circle of which the arc is a part.
See Radius of an Arc for ways to calculate the arc radius when you know other properties of the arc.
Central AngleCentral angle of an arcThe angle subtended by the arc to the center of the circle of which it is a part. This angle is always twice the peripheral angle (see below). SeeCentral Angle definition for more.
Inscribed AngleInscribed angle of an arcThe angle subtended by the arc to any point on the circumference of the circle of which it is a part. This angle is always half the central angle (see above). See Inscribed Angle of an Arc for more.

Khan Academy Probability Events and Outcomes 1


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