We are tasked with solving the given system of linear equations:

1. $y = \frac{3}{4}x + 1$
2. $y = -\frac{4}{3}x + 1$

Step 1: Set the two equations equal to each other

To find the intersection point (solution of the system), set the right-hand sides of the equations equal to each other:

$\frac{3}{4}x + 1 = -\frac{4}{3}x + 1$

Step 2: Eliminate the constant term

Subtract 1 from both sides:

$\frac{3}{4}x = -\frac{4}{3}x$

Step 3: Solve for $x$

To eliminate the fractions, find a common denominator. The least common denominator (LCD) of 4 and 3 is 12. Multiply through by 12:

$12 \cdot \frac{3}{4}x = 12 \cdot \left(-\frac{4}{3}x\right)$

$9x = -16x$

Combine like terms:

$9x + 16x = 0$

$25x = 0$

$x = 0$

Step 4: Solve for $y$

Substitute $x = 0$ into either equation to solve for $y$. Using the first equation:

$y = \frac{3}{4}(0) + 1$

$y = 1$

Final Solution:

The solution to the system of equations is:

$\boxed{(0, 1)}$

This means the lines intersect at the point $(0, 1)$.

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Questions for Exploration:

1. How do we know that this solution is unique?
2. What happens if the slopes of the lines are the same?
3. Can this method be extended to systems of three equations?
4. How does graphing help verify the solution?
5. Why is finding a common denominator important in solving equations with fractions?

Tip:

When dealing with systems of linear equations, graphing can provide a visual confirmation of the solution, but algebra ensures precision!