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Question 75071: Please help . And thanks in advance
Write a system of two linear equations that has .
a) only one solution ,(2,3).
b) an infinite number of solutions .
c) no solution.
Found 2 solutions by bucky, yawney2005:
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Write a system of two linear equations that has:
.
a) only one solution ,(2,3)
.
An easy way to do this one is to use the slope intercept form y = mx + b. where m is the slope
of the graph and b is the value where the graph crosses the y-axis.
For the first linear equation let's say the slope is -2. Then it's slope intercept form
becomes y = -2x + b. We can now solve for b because we know that the point (2,3) is on the
graph. Substitute 2 for x and 3 for y and the equation becomes:
.
3 = -2(2) + b
.
multiply on the right side and the equation then becomes:
.
3 = -4 + b
.
Solve for b by adding +4 to both sides to get rid of the -4 on the right side. When you
add + 4 to both sides the value of b computes to be:
.
7 = b
.
Substitute this value for b into the slope intercept form and you get:
.
y = -2x + 7
.
This is the first linear equation, and (2,3) is one of the points on its graph.
.
For the second linear equation assume a different value for the slope than you did last time.
Last time we let the slope be -2. Suppose this time we make the slope +2 so that we
know the graphs will not be parallel lines. The slope intercept form then becomes:
.
y = 2x + b
.
Again we can find b by substituting 3 for y and 2 for x. This leads to:
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3 = 2(2) + b
.
Multiply out the right side to get:
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3 = 4 + b
.
Solve for b by subtracting 4 from both sides to get:
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-1 = b
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Substitute this value for b into the slope intercept form we are working on, and you get:
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y = 2x - 1
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(2, 3) is a point on this graph. So now we have two equations:
.
y = -2x + 7 and
y = +2x - 1
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And both have the common solution of (2,3).
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b) an infinite number of solutions
All that you need to do here is to write an equation and make a second equation a multiple
of the first equation. Example:
.
The first equation is y = 2x -1
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The second equation is 2y = 4x - 2
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This second equation is just 2 times the first equation. That makes them the same equation
so they will have the same graph and every solution of the first equation also satisfies the
second equation. Therefore, these two equations have an infinite number of common
solutions.
.
c) no solution.
.
No solution will happen if the two graphs are parallel, but are a separate line. To make them
parallel we need only make sure that the linear graphs have the same slope. However, they
need to be separate lines, so we can make them cross the y-axis at different points.
.
Let's again use the slope intercept form. And let's make the slope of both of our lines +3.
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When we do, the equation becomes y = 3x + b
.
Now we can just substitute two different values for b, the point where the graph crosses the
y-axis. That forces the graphs to be separate lines. So we could write:
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y = 3x + 5 and
y = 3x - 2
.
These two graphs are parallel because they have the same slope of +3. Remember that parallel
lines never cross and to have a common solution, two linear equations must have graphs that
cross at a point, and the x and y values at that point are the common solution for both
equations. So we've made the equations have the same slope +3 making their graphs parallel
lines. However, the first equation has a graph that crosses the y-axis at + 5 and the second
has a graph that crosses the y-axis at -2. Therefore, the graphs of these two equations
are parallel but are separated by 7 vertical units (the algebraic difference between 5 and -2).
.
Hope this helps you to see the meaning of 1 common solution, an infinite number of
solutions, and no common solution and how they tie into the form of the equation and
the graphs of the two equations.
Answer by yawney2005(1) (Show Source):
You can put this solution on YOUR website! I am having problems as well with how to graph this problem y = 3/4x + 2. Can you help me solve this problem? On the Graphing linear equations
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