Determine the number of solutions to this system of linear equations: 4x+8y=12 3x+6y=9

Answer 1

To determine the number of solutions to the system of linear equations given by:

1. $4x+8y=12$
2. $3x+6y=9$

we start by analyzing both equations individually.

Step 1: Simplify both equations

Equation 1: $$4x+8y=12$$ We can divide the entire equation by 4: $$x+2y=3\text{(Equation 1 simplified)}$$

Equation 2: $$3x+6y=9$$ We can divide the entire equation by 3: $$x+2y=3\text{(Equation 2 simplified)}$$

Step 2: Compare the simplified equations

After simplifying, we obtain: $$x+2y=3\text{(from Equation 1)}$$ $$x+2y=3\text{(from Equation 2)}$$

Both equations are identical. Therefore, they represent the same line in the coordinate plane.

Conclusion

Since both equations are the same, there are infinitely many solutions to this system of equations. The solutions can be expressed in terms of one variable (e.g., $y$), where $x$ can be expressed as: $$x=3-2y$$ Thus, for every value of $y$, there corresponds a unique value of $x$.

The final answer is: $$\text{Infinitely many solutions.}$$