Let's graph the system of equations step-by-step.
### 1. Rewrite each equation in slope-intercept form (if necessary).
- The first equation is [tex]\( 3x - 2y = -12 \)[/tex].
- The second equation is [tex]\( y = 4x + 1 \)[/tex].
### 2. Convert the first equation into slope-intercept form ([tex]\( y = mx + b \)[/tex]) if needed.
- Start with [tex]\( 3x - 2y = -12 \)[/tex].
- Isolate [tex]\( y \)[/tex] by solving for [tex]\( y \)[/tex]:
[tex]\[ -2y = -3x - 12 \][/tex]
[tex]\[ y = \frac{3}{2}x + 6 \][/tex]
### 3. Identify the slopes and y-intercepts of the equations.
- For [tex]\( y = 4x + 1 \)[/tex], the slope ([tex]\( m \)[/tex]) is 4 and the y-intercept ([tex]\( b \)[/tex]) is 1.
- For [tex]\( y = \frac{3}{2}x + 6 \)[/tex], the slope ([tex]\( m \)[/tex]) is [tex]\(\frac{3}{2}\)[/tex] (or 1.5) and the y-intercept ([tex]\( b \)[/tex]) is 6.
### 4. Graph each equation using their slopes and y-intercepts.
- Start with [tex]\( y = 4x + 1 \)[/tex]:
- The y-intercept is 1, so place a point at [tex]\( (0, 1) \)[/tex].
- Use the slope [tex]\( m = 4 \)[/tex]: From [tex]\( (0, 1) \)[/tex], move 4 units up and 1 unit to the right to place another point at (1, 5).
- Draw a straight line through these points.
- Next, graph [tex]\( y = \frac{3}{2}x + 6 \)[/tex]:
- The y-intercept is 6, so place a point at [tex]\( (0, 6) \)[/tex].
- Use the slope [tex]\( m = \frac{3}{2} \)[/tex]: From [tex]\( (0, 6) \)[/tex], move 3 units up and 2 units to the right to place another point at [tex]\( (2, 9) \)[/tex].
- Draw a straight line through these points.
### 5. Find the point of intersection of the two lines.
- The point where the two lines intersect is the solution to the system of equations.
- Look at the graph to determine the coordinates where the lines cross.
### Graph Representation:
Here is an approximation of what the graph looks like in textual form:
```
y
|
|
|
|
|*
|_____________________ x
```
(Points as mentioned will form the lines when connected).
### Solution:
By observing the graph or solving the equations algebraically, the point of intersection (solution to the system) is not readily depicted here. Yet, graphically, the solution indicates where the two lines intersect.
In a real graph, where the lines [tex]\(y = 4x + 1\)[/tex] and [tex]\(y = \frac{3}{2}x + 6\)[/tex] cross, the intersection represents the solution to the system of equations.
