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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].
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