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Q: What is true about the lines represented by this system of linear equations?

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The system of equations can have zero solutions, one solution, two solutions, any finite number of solutions, or an infinite number of solutions. If it is a system of LINEAR equations, then the only possibilities are zero solutions, one solution, and an infinite number of solutions. With linear equations, think of each equation describing a straight line. The solution to the system of equations will be where these lines intersect (a point). If they do not intersect at all (or maybe two of the lines intersect, and the third one doesn't) then there is no solution. If the equations describe the same line, then there will be infinite solutions (every point on the line satisfies both equations). If the system of equations came from a real world problem (like solving for currents or voltages in different parts of a circuit) then there should be a solution, if the equations were chosen properly.

Two or more straight lines meeting at one point.

There must be fewer independent equation than there are variables. An equation in not independent if it is a linear combination of the others.

They are all lines. Their equations are written in the slope-intercept form, where we clearly can see if they just intersect, or are perpendicular to each other, or parallel, or coincide.

x+y=12 is a linear (straight line) equation in the xy-plane. Alternate forms are y=12-x and x=12-y. The form y=12-x is most familiar, and it is a straight line with a slope of -1 that intercepts the y-axis at 12 and the x-axis at 12.

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The coordinates of the point of intersection represents the solution to the linear equations.

A "system" of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.

The two equations represent parallel lines.

perpendicular

The set of points the graphed equations have in common. This is usually a single point but the lines can be coincident in which case the solution is a line or they can be parallel in which case there are no solutions to represent.

A system of equations will have no solutions if the line they represent are parallel. Remember that the solution of a system of equations is physically represented by the intersection point of the two lines. If the lines don't intersect (parallel) then there can be no solution.

They do not. A set of lines can also be considered as a system of linear equations. But the fact that there is such a system does not mean that the lines intersect.

If you refer to linear equations, graphed as straight lines, two inconsistent equations would result in two parallel lines.

one solution; the lines that represent the equations intersect an infinite number of solution; the lines coincide, or no solution; the lines are parallel

A system of linear equations is consistent if there is only one solution for the system. Thus, if you see that the drawn lines intersect, you can say that the system is consistent, and the point of intersection is the only solution for the system. A system of linear equations is inconsistent if it does not have any solution. Thus, if you see that the drawn lines are parallel, you can say that the system is inconsistent, and there is not any solution for the system.

For two linear equations, they are equations representing parallel lines. (The lines must not be concurrent because if they are, you will have an infinite number of solutions.) For example y = mx + b and y = mx + c where b and c are different numbers are two non-concurrent parallel lines. The equations have no solution. With more than two linear equations there is much more scope. Unless ALL the lines meet at one point, the system will not have a solution. So a system consisting of equations defining the three lines of a triangle, for example, will not have a solution.

The pair of equations have one ordered pair that is a solution to both equations. If graphed the two lines will cross once.

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