https://file.aiquickdraw.com/mathbot/file/17271467124991h256msk.jpg

To solve the system of linear equations given in the image, we need to rewrite both equations in slope-intercept form (\( y = mx + b \)) and then graph them. Here's the process:

### The equations provided:
\[
\frac{1}{5}x + \frac{9}{4}y = \frac{311}{20}
\]
\[
-\frac{7}{2}x - \frac{5}{2}y = -14
\]

### Step 1: Solve the first equation for \( y \)
Start with:
\[
\frac{1}{5}x + \frac{9}{4}y = \frac{311}{20}
\]
Subtract \(\frac{1}{5}x\) from both sides:
\[
\frac{9}{4}y = \frac{311}{20} - \frac{1}{5}x
\]
Convert \(\frac{1}{5}x\) to have the same denominator as \(\frac{311}{20}\):
\[
\frac{1}{5} = \frac{4}{20}, \quad \text{so} \quad \frac{1}{5}x = \frac{4}{20}x
\]
Now the equation is:
\[
\frac{9}{4}y = \frac{311}{20} - \frac{4}{20}x
\]
Simplify:
\[
\frac{9}{4}y = \frac{311 - 4x}{20}
\]
Multiply both sides by \(\frac{4}{9}\) to solve for \( y \):
\[
y = \frac{4}{9} \times \frac{311 - 4x}{20}
\]
Distribute:
\[
y = \frac{4(311 - 4x)}{180}
\]
Simplify further to get the slope-intercept form.

### Step 2: Solve the second equation for \( y \)
Start with:
\[
-\frac{7}{2}x - \frac{5}{2}y = -14
\]
Add \(\frac{7}{2}x\) to both sides:
\[
-\frac{5}{2}y = \frac{7}{2}x - 14
\]
Multiply both sides by \(-\frac{2}{5}\):
\[
y = -\frac{7}{5}x + \frac{28}{5}
\]

Now that both equations are in slope-intercept form, you can graph them and identify the point where the two lines intersect. This point will be the solution to the system.

Do you want a more detailed breakdown or any specific clarifications?

Here are 5 relative questions for further understanding:
1. What does the slope represent in the slope-intercept form of a line?
2. How do you graph a line using the slope and y-intercept?
3. What are the possible outcomes when solving a system of linear equations graphically?
4. How can you confirm if two lines are parallel by looking at their slopes?
5. What is the geometric meaning of the solution to a system of two linear equations?

**Tip:** Always check if the equations need to be simplified before solving them to make graphing easier!

Solve the following system of linear equations by graphing: 1/5x + 9/4y = 311/20 and -7/2x - 5/2y = -14

Learn how to solve the system of linear equations 1/5x + 9/4y = 311/20 and -7/2x - 5/2y = -14 by graphing. This involves converting both equations to slope-intercept form and identifying the point of intersection.

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