To solve the system of linear equations using the graphing method, let's break it down step by step:

Equations:

1. $3x + y = 4$
2. $\frac{9}{2}x + \frac{3}{2}y = 6$

We need to find the x- and y-intercepts of both lines.

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Step 1: Find the x- and y-intercepts for Line 1: $3x + y = 4$

To find the x-intercept:

Set $y = 0$ in the equation and solve for $x$:

$3x + 0 = 4$ $3x = 4$ $x = \frac{4}{3}$ So, the x-intercept for Line 1 is $\left( \frac{4}{3}, 0 \right)$.

To find the y-intercept:

Set $x = 0$ in the equation and solve for $y$:

$3(0) + y = 4$ $y = 4$ So, the y-intercept for Line 1 is $(0, 4)$.

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Step 2: Find the x- and y-intercepts for Line 2: $\frac{9}{2}x + \frac{3}{2}y = 6$

To find the x-intercept:

Set $y = 0$ in the equation and solve for $x$:

$\frac{9}{2}x + \frac{3}{2}(0) = 6$ $\frac{9}{2}x = 6$ $x = \frac{6 \times 2}{9} = \frac{12}{9} = \frac{4}{3}$ So, the x-intercept for Line 2 is $\left( \frac{4}{3}, 0 \right)$.

To find the y-intercept:

Set $x = 0$ in the equation and solve for $y$:

$\frac{9}{2}(0) + \frac{3}{2}y = 6$ $\frac{3}{2}y = 6$ $y = \frac{6 \times 2}{3} = 4$ So, the y-intercept for Line 2 is $(0, 4)$.

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Step 3: Graph the lines

Both lines have the same intercepts:

• x-intercept: $\left( \frac{4}{3}, 0 \right)$
• y-intercept: $(0, 4)$

This means that the two lines coincide and are dependent, which means they represent the same line.

Step 4: Final Solution

Since the lines are identical (dependent), the solution is the entire line, which can be expressed as:

$\boxed{(x, y) = (t, 4 - 3t)} \quad \text{for any real number} \, t.$

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Chart of Intercepts:

Line	x-intercept	y-intercept
Line 1	$\left( \frac{4}{3}, 0 \right)$	$(0, 4)$
Line 2	$\left( \frac{4}{3}, 0 \right)$	$(0, 4)$

Tip:

When you encounter two lines that have identical intercepts and are thus dependent, the system has an infinite number of solutions because every point on the line is a solution to the system.

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Let me know if you'd like further clarification or if you have any other questions!

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Here are some related questions to deepen your understanding:

1. How do you find the slope of a line from its equation in standard form?
2. How do you interpret the solution of a dependent system in geometric terms?
3. What happens if two lines have different slopes but the same intercepts?
4. How can you check if two lines are parallel or perpendicular from their equations?
5. What does it mean for a system of equations to be inconsistent?