isabella292008
Answered

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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
To determine the solution to the given system of equations, we need to find the point at which the two lines intersect. Here are the equations:

1. [tex]\( y = 3x + 9 \)[/tex]
2. [tex]\( 6x + 2y = 6 \)[/tex]

### Step-by-Step Solution

#### Step 1: Convert the Second Equation to Slope-Intercept Form
The second equation is [tex]\( 6x + 2y = 6 \)[/tex]. To express this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], follow these steps:

1. Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ 2y = 6 - 6x \][/tex]
2. Divide every term by 2:
[tex]\[ y = -3x + 3 \][/tex]

Now, we have the two equations in slope-intercept form:
- [tex]\( y = 3x + 9 \)[/tex]
- [tex]\( y = -3x + 3 \)[/tex]

#### Step 2: Find the Intersection Point
To find the point of intersection, set the right-hand sides of the equations equal to each other:

[tex]\[ 3x + 9 = -3x + 3 \][/tex]

Solve for [tex]\( x \)[/tex]:

1. Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[ 6x + 9 = 3 \][/tex]
2. Subtract 9 from both sides:
[tex]\[ 6x = -6 \][/tex]
3. Divide by 6:
[tex]\[ x = -1 \][/tex]

With [tex]\( x = -1 \)[/tex], substitute this value into one of the original equations to find [tex]\( y \)[/tex]. Using [tex]\( y = 3x + 9 \)[/tex]:

[tex]\[ y = 3(-1) + 9 = -3 + 9 = 6 \][/tex]

Therefore, the solution to the system is [tex]\( (-1, 6) \)[/tex].

### Conclusion
The system of equations has one unique solution: [tex]\( (-1, 6) \)[/tex].

The graphical representations of these equations will intersect at this point. Consequently, the correct answer is:

There is one unique solution [tex]\((-1, 6)\)[/tex].
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