The need for a one-of-a-kind solution to linear equations If the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 cross at a location, a system of linear equations ax + by + c = 0 will have a unique solution. Specifically, if the two lines are not parallel or coincident.

Any number of pairs of solutions can be found for any set of conditions that does not include parallel or coincident lines. For example, if the lines are parallel, there is only one solution; if they pass through the same point, there are infinitely many solutions.

Students often think that because there can be as many solutions as there are variables in the equation that it is impossible to solve for anything. But this is not true - you just have to know how to use what is called the "linear algebra" toolkit to solve **these problems**. The key idea is that if you assume that there is at least one solution, then you can prove that it must be unique. This means that there can be at **most one solution** for **any given set** of conditions.

Here's an example problem that uses all of **these concepts**. Suppose that the lines ax + by + c = 0 and dx + ey + g = 0 intersect at the point (x,y).

- What is the unique solution to the linear equation?
- What are the solutions to two linear equations in two variables if they are parallel lines?
- Does the pair of the linear equation have no solution?
- Why can’t a linear system have two solutions?
- What does it mean when a system of two linear equations has no solution?
- Do systems of equations have one solution?

There is no solution to the pair of linear equations if the lines are parallel. The equations are said to be inconsistent if there is no solution to the provided pair of linear equations. There would be no way to determine which equation should be satisfied by an unknown value unless one of them was known or assumed.

Two linear equations can be consistent even if they don't have a solution. A simple example is when the two equations have equal coefficients but different slopes. In **this case**, there will be two values of the variable that will make the equations true and they won't have any solution. However, sometimes people talk about **these cases** where the equations don't have a solution even though there are real numbers that satisfy both equations. These cases are called inconsistent pairs and they tell us something important about the given set of equations: They cannot all be satisfied at the same time.

In general, there will be infinitely many pairs of values that will make the equations false. We can only hope that somewhere among them there will be a pair that satisfies **both equations**. If we could find **such a pair**, then the equations would have a solution after all.

People usually look for solutions by trying values for the variable.

As a result, neither equation has a solution. There is no solution if two lines are parallel. If two lines are coinciding, they have an infinite solution and the pair of linear equations is consistent. As a result, **the two lines** overlap at a point (a, b). This point is called a common point of the two lines.

If two parallel (and non-coincident) lines do not overlap, there is no solution. If two lines are coplanar and non-parallel, they will collide at **one place** only. That is the only solution. They will not intersect if they are not coplanar, and so there will be no solution.

A line has exactly two directions in which it can be extended: forward or backward. If it were possible to extend it in **both directions** simultaneously, this would mean that the line itself was not unique; any point on it could serve as its origin. But since all lines are uniquely determined by where they start, this cannot be the case.

A line has an infinite number of points at which it can be divided into two parts, each going in **a different direction**. These points of division are called extensions of the line and the directions in which it extends are called branches of the line. A branch may be thought of as **a small copy** of the line starting at **every point** on it.

Because lines have an infinite number of extensions, they can never end up with more than one extension. If they did, they would contain a circle as their set of all points, which is impossible because circles are bounded sets.

Lines are very useful for describing relationships between objects because they can show whether or not these objects overlap. If two lines do not overlap, then they are called parallel.

A two-linear equation system might have one solution, an unlimited number of solutions, or no solution at all. When a system does not have a solution, it is said to be inconsistent. Because the line graphs do not overlap, the graphs are parallel, thus there is no solution.

If you add 3+4=7 to 2x+3=5 then the answer is not 7. Instead, you get 3+6=9 which is not equal to either of the original numbers. This shows that your original equation system was inconsistent. There is no way for it to have a single solution - because if there were, then these would be the same number, which they aren't.

In mathematics, a system of equations describes a relationship between a set of variables. The simplest example is a linear equation system, where the variables are separated into groups with each group containing an equal number of variables. For example, given the equation systems x + y = z and 2x + 3y = 4z, we can divide the variables into pairs by putting every term with a single variable in its own column: x/y = z/0 and 2x/y = 4z/0. Since each column contains an equal number of variables, this system of equations is called balanced. It has only one solution since any multiplication of values from different columns gives **a new column** whose entries are also multiplied together.

A linear equation system normally has a single solution, but it can also have no solution (parallel lines) or infinite solutions (same line). This page discusses all three situations. One possible solution When the graphs of a system of **linear equations** cross at a location, there is only one solution. For example, if the graph of **the first equation** in the list below crosses the y-axis at (3, 4), then there is only one value of x for which the statement holds true, and that value is 3. Similarly, if the graph of the third equation in the list crosses the y-axis at (4, 5), then there is only one value of x for which the statement holds true, and that value is 4. By contrast, if the graph of the second equation in the list crossed the y-axis at (4, 4), there would be two values of x for which the statement held true.

To see this, consider what would happen if there were multiple values of x for which the statement was true. Then the line through those points would intersect the y-axis more than once, which cannot happen with these equations because they all involve arithmetic operations on real numbers. Therefore, there is only one value of x for which the statement is true, and that proves that the system of equations has only one solution.