Graphical Method for Linear Programming Problems - Videos - Free Printable

The image shows two lines on a coordinate plane, each representing a linear equation:

1. Blue line: $ 8.8x + 5.2y = 60 $
2. Red line: $ 8.8x + 5.2y = 20 $

We are asked to analyze or solve the problem related to these two equations.

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Step 1: Understand the Equations

Both equations have the same left-hand side (LHS):
$ 8.8x + 5.2y $, but different right-hand sides (RHS): 60 and 20.

This means both are of the form:
$$
8.8x + 5.2y = C
$$
where $ C $ is a constant.

These represent parallel lines, because they have the same slope but different intercepts.

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Step 2: Find Slope

Let’s rewrite one of the equations in slope-intercept form ($ y = mx + b $) to find the slope.

Start with:
$$
8.8x + 5.2y = 60
$$

Solve for $ y $:
$$
5.2y = -8.8x + 60
$$
$$
y = \frac{-8.8}{5.2}x + \frac{60}{5.2}
$$

Simplify:
- $ \frac{-8.8}{5.2} = \frac{-88}{52} = \frac{-22}{13} \approx -1.6923 $
- $ \frac{60}{5.2} = \frac{600}{52} = \frac{150}{13} \approx 11.538 $

So the blue line has equation:
$$
y = -\frac{22}{13}x + \frac{150}{13}
$$

Similarly, for the red line:
$$
8.8x + 5.2y = 20
\Rightarrow 5.2y = -8.8x + 20
\Rightarrow y = -\frac{8.8}{5.2}x + \frac{20}{5.2}
= -\frac{22}{13}x + \frac{50}{13}
$$

So both lines have the same slope $ m = -\frac{22}{13} $, confirming they are parallel.

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Step 3: Interpretation

Since the two lines are parallel and not identical (different constants), they never intersect.

Therefore, the system of equations:
$$
\begin{cases}
8.8x + 5.2y = 60 \\
8.8x + 5.2y = 20
\end{cases}
$$
has no solution — it's inconsistent.

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Step 4: Visual Confirmation

From the graph:
- The blue line intersects the y-axis around $ y = 10 $ (since at $ x = 0 $, $ 5.2y = 60 \Rightarrow y \approx 11.54 $)
- The red line intersects the y-axis around $ y = 4 $ (since $ 5.2y = 20 \Rightarrow y \approx 3.85 $)

They are clearly parallel and separated vertically.

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Final Answer:

✔ The two lines are parallel and do not intersect.

✘ There is no solution to the system of equations.

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Summary:

- Equations: $ 8.8x + 5.2y = 60 $ and $ 8.8x + 5.2y = 20 $
- Relationship: Parallel lines (same slope, different intercepts)
- Conclusion: No solution exists — the system is inconsistent.

Let me know if you'd like to explore what happens if the RHS were equal or if you're solving for something else (e.g., distance between lines).