Sure, let's start by determining the equations of the lines from the given tables and then find the intersection point.
### Step 1: Finding the first equation
We are given points from the first table:
[tex]\[ (-1, 1), (0, 3), (1, 5), (2, 7) \][/tex]
To find the slope (m) of the first equation, we can use two points, say [tex]\((0, 3)\)[/tex] and [tex]\((1, 5)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{1 - 0} = \frac{2}{1} = 2 \][/tex]
So, the slope is [tex]\(2\)[/tex].
Given that the y-intercept (b) can be directly read from the point [tex]\((0, 3)\)[/tex], we have:
[tex]\[ b = 3 \][/tex]
Thus, the first equation is:
[tex]\[ y = 2x + 3 \][/tex]
### Step 2: Finding the second equation
We are given points from the second table:
[tex]\[ (-2, -7), (0, -1), (2, 5), (4, 11) \][/tex]
To find the slope (m) of the second equation, we can use two points, say [tex]\((0, -1)\)[/tex] and [tex]\((2, 5)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{2 - 0} = \frac{5 + 1}{2} = \frac{6}{2} = 3 \][/tex]
So, the slope is [tex]\(3\)[/tex].
The y-intercept (b) can be directly read from the point [tex]\((0, -1)\)[/tex], but we need to correctly arrange it as per the format given in the question:
[tex]\[ y = 3x - b \][/tex]
[tex]\[ -1 = 3(0) - b \][/tex]
[tex]\[ -1 = -b \][/tex]
[tex]\[ b = 1 \][/tex]
Thus, the second equation is:
[tex]\[ y = 3x - 1 \][/tex]
### Step 3: Finding the intersection point
To find the intersection, we set the two equations equal:
[tex]\[ 2x + 3 = 3x - 1 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 2x + 3 = 3x - 1 \][/tex]
[tex]\[ 3 + 1 = 3x - 2x \][/tex]
[tex]\[ 4 = x \][/tex]
Then substituting [tex]\(x = 4\)[/tex] back into one of the equations to find [tex]\(y\)[/tex]:
[tex]\[ y = 2(4) + 3 = 8 + 3 = 11 \][/tex]
Thus, the solution of the system is:
[tex]\[ (4, 11) \][/tex]
Therefore, the completed statements are:
- The first equation of this system is [tex]\( y = 2x + 3 \)[/tex]
- The second equation of this system is [tex]\( y = 3x - 1 \)[/tex]
- The solution of the system is [tex]\( (4, 11) \)[/tex]
So, the final answers are:
- The first equation of this system is [tex]\( y = 2x + 3 \)[/tex]
- The second equation of this system is [tex]\( y = 3x - 1 \)[/tex]
- The solution of the system is [tex]\( (4, 11) \)[/tex]
