Sure, let's break down the process of graphing the system of equations and finding the solution step by step.
We are given the system of equations:
[tex]\[
\begin{array}{l}
y = 3x + 9 \\
6x + 2y = 6
\end{array}
\][/tex]
### Step 1: Graph the First Equation
The first equation is:
[tex]\[
y = 3x + 9
\][/tex]
This is a linear equation in slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
For [tex]\(y = 3x + 9\)[/tex]:
- The slope [tex]\(m = 3\)[/tex]
- The y-intercept [tex]\(b = 9\)[/tex]
To graph this equation, start at the point (0, 9) on the y-axis and use the slope to find another point. The slope of 3 means you rise 3 units for every 1 unit you run to the right. So, starting at (0, 9):
- When [tex]\(x = 1\)[/tex], [tex]\(y = 3(1) + 9 = 12\)[/tex]. (Point is (1, 12))
- When [tex]\(x = -1\)[/tex], [tex]\(y = 3(-1) + 9 = 6\)[/tex]. (Point is (-1, 6))
Draw a line through these points.
### Step 2: Graph the Second Equation
The second equation is:
[tex]\[
6x + 2y = 6
\][/tex]
First, we convert it to slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Divide the entire equation by 2 to simplify it:
[tex]\[
3x + y = 3
\][/tex]
- Solve for [tex]\(y\)[/tex]:
[tex]\[
y = -3x + 3
\][/tex]
Now, this is also a linear equation in slope-intercept form with:
- The slope [tex]\(m = -3\)[/tex]
- The y-intercept [tex]\(b = 3\)[/tex]
To graph this equation, start at the point (0, 3) on the y-axis and use the slope to find another point. The slope of -3 means you drop 3 units for every 1 unit you run to the right. So, starting at (0, 3):
- When [tex]\(x = 1\)[/tex], [tex]\(y = -3(1) + 3 = 0\)[/tex]. (Point is (1, 0))
- When [tex]\(x = -1\)[/tex], [tex]\(y = -3(-1) + 3 = 6\)[/tex]. (Point is (-1, 6))
Draw a line through these points.
### Step 3: Determine the Intersection
By graphing both lines, we see where they intersect.
The lines intersect at the point [tex]\((-1, 6)\)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
(-1, 6)
\][/tex]
So, the correct answer is:
- There is one unique solution [tex]\( (-1, 6) \)[/tex].
