Find all solutions of the given system of equations using elimination. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of t, where y = y(t).)
4x − 3y	 = 	5
12x − 9y	 = 	15


We are given the system of equations:

\[
4x - 3y = 5 \tag{1}
\]
\[
12x - 9y = 15 \tag{2}
\]

To solve this system using elimination, let's first notice that the second equation is simply a multiple of the first equation. Specifically, the second equation can be obtained by multiplying the first equation by 3:

\[
3(4x - 3y) = 3(5)
\]
\[
12x - 9y = 15 \tag{3}
\]

This shows that equations (2) and (3) are identical, meaning the two equations represent the same line. As a result, this system of equations is **dependent**, meaning there are infinitely many solutions.

To express the solution, we solve one of the equations for \(x\) or \(y\). Let's solve the first equation for \(x\) in terms of \(y\).

From the first equation:
\[
4x - 3y = 5
\]
Solve for \(x\):
\[
4x = 3y + 5
\]
\[
x = \frac{3y + 5}{4}
\]

Thus, the solution can be written in terms of \(y\), where \(y = t\) (a parameter):
\[
x = \frac{3t + 5}{4}, \quad y = t
\]

Therefore, the general solution is:
\[
(x, y) = \left(\frac{3t + 5}{4}, t\right)
\]
where \(t\) is any real number.

Would you like more details on the elimination process or parametric solutions? Here are some related questions:

1. How can you verify if a system of equations is dependent?
2. What is the difference between dependent and inconsistent systems?
3. How can substitution be used to solve the same system of equations?
4. What happens when you graph dependent systems of equations?
5. How does parameter \(t\) represent the solution set geometrically?

**Tip**: A system is dependent if one equation is a scalar multiple of the other, meaning they represent the same line.

Find all solutions of the given system of equations using elimination. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of t, where y = y(t).) 4x − 3y = 5 12x − 9y = 15

Discover how to solve the system of linear equations 4x − 3y = 5 and 12x − 9y = 15 using the elimination method. This problem involves recognizing dependent systems, scalar multiples, and solving for x in terms of y. Ideal for high school students (Grades 9-12).

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