Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Graph the following system of equations:

[tex]\[ \begin{array}{l}
y = 3x + 9 \\
6x + 2y = 6
\end{array} \][/tex]

What is the solution to the system?

A. There is no solution.
B. There is one unique solution [tex]\((-1, 6)\)[/tex].
C. There is one unique solution [tex]\((0, 3)\)[/tex].
D. There are infinitely many solutions.

Sagot :

GinnyAnswer
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].
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.