To solve the system of equations graphically and algebraically, we can follow these steps:
1. Graph the first equation:
[tex]\[ y = 3x + 9 \][/tex]
The equation [tex]\( y = 3x + 9 \)[/tex] is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = 3 \)[/tex] is the slope and [tex]\( b = 9 \)[/tex] is the y-intercept.
- To plot this line, start by locating the y-intercept (0, 9) on the graph.
- From the y-intercept, use the slope. Since the slope is 3, which means "rise over run" is 3/1, move up 3 units and 1 unit to the right to find another point on the line. Repeat this process to draw the line.
2. Graph the second equation:
[tex]\[ 6x + 2y = 6 \][/tex]
To write this in slope-intercept form [tex]\( y = mx + b \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[ 2y = -6x + 6 \][/tex]
[tex]\[ y = -3x + 3 \][/tex]
This second linear equation, [tex]\( y = -3x + 3 \)[/tex], has a slope [tex]\( m = -3 \)[/tex] and a y-intercept [tex]\( b = 3 \)[/tex].
- To plot this line, start by locating the y-intercept (0, 3) on the graph.
- From the y-intercept, use the slope. Since the slope is -3, move down 3 units and 1 unit to the right, or equivalently, move up 3 units and 1 unit to the left to find another point on the line. Repeat this process to draw the line.
3. Find the intersection point:
The solution to the system of equations is the point where the two lines intersect.
4. Solving algebraically:
Let's solve the system of equations algebraically to confirm the graphical intersection point.
[tex]\[
y = 3x + 9 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
6x + 2y = 6 \quad \text{(Equation 2)}
\][/tex]
Substitute [tex]\( y \)[/tex] from Equation 1 into Equation 2:
[tex]\[
6x + 2(3x + 9) = 6
\][/tex]
Simplify:
[tex]\[
6x + 6x + 18 = 6
\][/tex]
[tex]\[
12x + 18 = 6
\][/tex]
Subtract 18 from both sides:
[tex]\[
12x = -12
\][/tex]
Divide by 12:
[tex]\[
x = -1
\][/tex]
Substitute [tex]\( x = -1 \)[/tex] back into Equation 1 to find [tex]\( y \)[/tex]:
[tex]\[
y = 3(-1) + 9
\][/tex]
[tex]\[
y = -3 + 9
\][/tex]
[tex]\[
y = 6
\][/tex]
Thus, the solution to the system is the point [tex]\( (-1, 6) \)[/tex].
Therefore, the correct answer to the question is:
There is one unique solution [tex]\((-1,6)\)[/tex].
