We are given the system of linear equations:

-10x + 14y = 12 \tag{1} 5x - 7y = -6 \tag{2}

Step 1: Eliminate one variable

We can use the elimination method by adding or subtracting the equations after multiplying them by a suitable factor to eliminate one variable.

Notice that equation (2) is already the same as equation (1) but with opposite signs. If we multiply equation (2) by 2, we can directly add the two equations to eliminate one variable.

Multiplying equation (2) by 2:

$2(5x - 7y) = 2(-6)$ 10x - 14y = -12 \tag{3}

Now, add equation (1) and equation (3):

$(-10x + 14y) + (10x - 14y) = 12 + (-12)$ $0 = 0$

Step 2: Analyze the result

The result $0 = 0$ is a true statement, indicating that the system of equations is dependent. This means that the two equations represent the same line, and therefore, there are infinitely many solutions.

Step 3: Parametric form

Since the system has infinitely many solutions, we need to express the solution in parametric form. We can express either $x$ or $y$ in terms of a parameter. Let's solve for $x$ in terms of $y$.

From equation (2):

$5x - 7y = -6$

Solve for $x$:

$5x = 7y - 6$ $x = \frac{7y - 6}{5}$

Thus, the parametric solution is:

$x = \frac{7y - 6}{5}, \quad y = t \quad (\text{where } t \text{ is a parameter})$

Final Answer:

The parametric solution is: $x = \frac{7t - 6}{5}, \quad y = t$

Would you like further details on any step?

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Relative Questions:

1. How can you identify if a system of linear equations is dependent or inconsistent?
2. What is the difference between a parametric solution and a unique solution?
3. How do you graphically interpret a system with infinitely many solutions?
4. Can you find the solution using the substitution method instead of elimination?
5. What is the significance of the parameter $t$ in a parametric solution?

Tip: When solving systems of equations, always check for cases of dependency or inconsistency by simplifying the equations.