Examples of solution in the following topics:
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- Some linear systems may not have a solution, while others may have an
infinite number of solutions.
- Even so, this does not guarantee a unique solution.
- A solution to the system above is given by
- An inconsistent system has no solution.
- A dependent system
has infinitely many solutions.
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- Equations in two variables have not one solution but a series of solutions that will satisfy the equation for both variables.
- Each solution is an ordered pair and can be written in the form $(x, y)$.
- For example, $(1, 2)$ is a solution to the equation.
- Another solution is $(30, 60)$, because $(60) = 2(30)$.
- Therefore, the solution is $(3, 5)$.
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- In our study of linear equations in two variables, we observed that all
the solutions to an equation—and only those solutions—
were located on the graph of that equation.
- The resulting ordered pair will be one solution to the equation.
- So, let's substitute $x = 0 $ to find one solution:
- Graph showing all possible solutions of the given inequality.
- The solutions lie in the shaded region, including the boundary line.
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- Independent systems have a single solution.
- Dependent systems have an infinite number of solutions.
- Inconsistent systems have no solution.
- An infinite number of
solutions can result from several situations.
- The three planes could be
the same, so that a solution to one equation will be the solution to
the other two equations.
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- Two properties of a linear system are consistency (are there solutions?
- A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied.
- A solution to the system above is given by
- A linear system is consistent if it has a solution, and inconsistent otherwise.
- Note that any two of these equations have a common solution.
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- For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
- An independent system of equations has exactly one solution $(x,y)$.
- An inconsistent system has no solution, and a dependent system has an infinite number of solutions.
- Note that there are an infinite number of solutions to a dependent system, and these solutions fall on the shared line.
- A
linear system is consistent if it has a solution, and inconsistent
otherwise.
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- To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.
- These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
- This also means that if there are inequalities that don't overlap, then there is no solution to the system.
- There are no solutions above the line.
- The origin is a solution to the system, but the point $(3,0)$ is not.
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- In an equation with one variable, the variable has a solution, or value, that makes the equation true.
- The values of the variables that make an equation true are called the solutions of the equation.
- Thus, we can easily check whether a number is a genuine solution to a given equation.
- For example, let's examine whether $x=3$ is a solution to the equation $2x + 31 = 37$.
- Therefore, we can conclude that $x = 3$ is, in fact, a solution to the equation $2x+31=37$.
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- Note that the expression x > 12 has infinitely many solutions.
- Some solutions are: 13, 15, 90, 12.1, 16.3, and 102.51.
- A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution.
- Speculate on the number of solutions of a linear inequality.
- A linear inequality may have infinitely many solutions or no solutions.
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- Thus if we find a solution $g(x)$ for our problem, we have actually found infinitely many solutions $cg(x)$, one for every non-zero number $c$.
- Thus for given zeros $x_1, x_2, \ldots, x_n$ we find infinitely many solutions
- Given zeros $0$, $1$, and $2$, our general solution is of the form
- In the picture below, the blue graph represents the solution for $c$ equal to $1$.
- The red graph represents the solution for $c$ equal to $-1/2$.