The tables of ordered pairs represent some points on the graphs of lines q and v.

Question

x y x y
-3 -3.5 -5 18
1 2.5 -1 10
4 7 3 -2
8 13 7 -6
what is the solution to the equation?
a.(1,6)
b.(2,-4)
c. (2,0)
d. (2,4)

Answer 1

To find the solution to the equations represented by the lines q and v based on the table of ordered pairs, we first need to determine the equations of both lines from the points given.

Line q has points:

1. (-3, -3.5)
2. (1, 2.5)
3. (4, 7)
4. (8, 13)

We will calculate the slope (m) of line q using two points, say (1, 2.5) and (4, 7):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2.5}{4 - 1} = \frac{4.5}{3} = \frac{3}{2} \]

Now we can use one point to find the y-intercept (b) using the slope-intercept form \(y = mx + b\).

Using point (1, 2.5): \[ 2.5 = \frac{3}{2}(1) + b \ b = 2.5 - \frac{3}{2} = 2.5 - 1.5 = 1 \]

So the equation of line q is: \[ y = \frac{3}{2}x + 1 \]

Line v has points:

1. (-5, 18)
2. (-1, 10)
3. (3, -2)
4. (7, -6)

We will calculate the slope of line v using two points, (1, 10) and (3, -2):

\[ m = \frac{-2 - 10}{3 - (-1)} = \frac{-12}{4} = -3 \]

Now we will use any point to find the y-intercept (b). Using point (-1, 10): \[ 10 = -3(-1) + b \ b = 10 + 3 = 13 \]

So the equation of line v is: \[ y = -3x + 13 \]

To find the solution to the system of equations for the lines (where they intersect), we set the equations equal to each other:

\[ \frac{3}{2}x + 1 = -3x + 13 \]

Now we can solve for \(x\):

1. Multiply by 2 to eliminate the fraction: \[ 3x + 2 = -6x + 26 \]
2. Add \(6x\) to both sides: \[ 9x + 2 = 26 \]
3. Subtract 2 from both sides: \[ 9x = 24 \]
4. Divide by 9: \[ x = \frac{24}{9} = \frac{8}{3} \]

Now substitute \(x = \frac{8}{3}\) back into one of the equations to find \(y\). Using line q's equation: \[ y = \frac{3}{2}\left(\frac{8}{3}\right) + 1 = 4 + 1 = 5 \]

The solution, in ordered pair form, is \(\left( \frac{8}{3}, 5 \right)\).

Among the answer options provided: a. (1,6) b. (2,-4) c. (2,0) d. (2,4)

There seems to be no option that corresponds exactly to \(\left( \frac{8}{3}, 5 \right)\). However, you can check if any of these options satisfy either of the line equations for a numerical match. If they do not, it would imply that there is no valid intersection point among the provided options.

Please let me know if you require further assistance!