The Relationship Between Coincident Lines and Infinitely Many Solutions in a System of Linear Equations

In a system of two linear equations, if two linear equations have the same graph, then the associated system has infinitely many solutions.

In a system of two linear equations, if the two equations have the same graph, it means that the lines representing the equations are coincident or overlap each other perfectly

In a system of two linear equations, if the two equations have the same graph, it means that the lines representing the equations are coincident or overlap each other perfectly.

When two lines are coincident, it implies that they have the same slope, and their y-intercepts are also the same. This indicates that both equations can be written in the form y = mx + b, where m represents slope and b represents the y-intercept.

Let’s consider the equations as y = mx + b₁ and y = mx + b₂, where m represents slope and b₁ and b₂ represent the y-intercepts.

Since the lines representing the equations are the same, both equations have the same slope (m) and the same y-intercept (b).

Now, when we solve this system of equations, we get:

y = mx + b₁
y = mx + b₂

Since both equations are the same, we can rewrite it as:

y = mx + b₂

So, we have reduced the system of equations to a single equation. This implies that any value of x will give us the same value of y, and therefore, we have infinitely many solutions.

In conclusion, if two linear equations have the same graph, the associated system has infinitely many solutions. This happens because the equations are dependent on each other and can be expressed as a single equation.

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