To solve the system of equations by graphing, we first interpret the equations provided:

1. $y = 3 - 2x$
2. $-4x - 4y = -20$

Step 1: Rewrite the second equation in slope-intercept form

We need to express the second equation ($-4x - 4y = -20$) in the form $y = mx + b$:

$-4x - 4y = -20$

Divide through by $-4$:

$x + y = 5$

Now isolate $y$:

$y = -x + 5$

Step 2: Graph the equations

• For $y = 3 - 2x$:

  • The slope is $-2$, and the y-intercept is $3$ ($0, 3$).
  • Plot the intercept, then use the slope to plot another point. For example:
    • When $x = 1$, $y = 3 - 2(1) = 1$ ($1, 1$).
  • Connect the points to draw the line.

• For $y = -x + 5$:

  • The slope is $-1$, and the y-intercept is $5$ ($0, 5$).
  • Plot the intercept, then use the slope to plot another point. For example:
    • When $x = 2$, $y = -2 + 5 = 3$ ($2, 3$).
  • Connect the points to draw the line.

Step 3: Find the point of intersection

The solution to the system is where the two lines intersect. Solving algebraically, we equate the two equations:

$3 - 2x = -x + 5$

Simplify:

$3 - 2x + x = 5$

$3 - x = 5$

$-x = 2$

$x = -2$

Substitute $x = -2$ into one of the original equations (e.g., $y = 3 - 2x$):

$y = 3 - 2(-2) = 3 + 4 = 7$

Thus, the point of intersection is $(-2, 7)$.

Final Answer:

The solution to the system is $(-2, 7)$.

Would you like me to provide a detailed graph or clarify any part of this solution?

Related Questions:

1. What does the slope of a line represent, and how does it help in graphing?
2. How can you verify the solution to a system of equations?
3. Why is slope-intercept form useful for graphing?
4. What are some alternative methods for solving systems of equations?
5. How can systems of equations be applied to real-world problems?

Tip: Always check your solution by substituting the intersection point into both original equations.