generate me a graph using the slope formula and explain how to use the slope formula for the following equations: 3x+42y = 87 and 28x - 12y = 4

To use the slope formula for the given equations, let's follow these steps:

### 1. Rearranging the equations into the slope-intercept form \(y = mx + b\), where \(m\) is the slope:

#### Equation 1: \( 3x + 42y = 87 \)
Rearranging:
\[
42y = -3x + 87 \quad \text{(move \(3x\) to the right side)}
\]
\[
y = \frac{-3}{42}x + \frac{87}{42} = -\frac{1}{14}x + \frac{29}{14}
\]
So, the slope \(m_1 = -\frac{1}{14}\).

#### Equation 2: \( 28x - 12y = 4 \)
Rearranging:
\[
-12y = -28x + 4 \quad \text{(move \(28x\) to the right side)}
\]
\[
y = \frac{28}{12}x - \frac{4}{12} = \frac{7}{3}x - \frac{1}{3}
\]
So, the slope \(m_2 = \frac{7}{3}\).

### 2. Slope Formula and Graph
The slope formula helps you identify how steep a line is, where:
\[
\text{slope} = m = \frac{\Delta y}{\Delta x}
\]
For these two lines:
- The first equation has a slope of \( -\frac{1}{14} \), meaning the line decreases slowly.
- The second equation has a slope of \( \frac{7}{3} \), meaning the line increases steeply.

The graph above visually shows these lines. The blue line represents the first equation, and the red line represents the second equation.

Would you like further details or have any questions?

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### 5 Related Questions:
1. How do you interpret the slope in a real-world context?
2. What does it mean when two lines have different slopes?
3. How would you find the point where these two lines intersect?
4. Can you solve these equations algebraically to find their intersection?
5. How does changing the slope affect the angle of a line?

### Tip:
Remember, a positive slope means the line goes upwards as \(x\) increases, while a negative slope means it goes downwards.

This guide explains how to use the slope formula and rearrange linear equations to graph them. Learn step-by-step how to convert 3x+42y=87 and 28x-12y=4 into slope-intercept form and identify their slopes for graphing. Suitable for Grades 8-10.

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