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24 The tables of ordered pairs represent some points on the graphs of lines
f and
g.
Line
f

x

y

2
7

4
10.5

7
15.75

11
22.75

Line
g

x

y

-3
4

-2
0

1
-12

4
-24

Which system of equations represents lines
f and
g ?
F

{:[y=1.75 x+3.5],[y=-4x-8]:}
G

{:[y=1.75 x+3.5],[y=-4x-2]:}
H

{:[y=3.5 x+1.75],[y=-4x-8]:}
J

{:[y=3.5 x+1.75],[y=-4x-2]:}

$24$ The tables of ordered pairs represent some points on the graphs of lines $f$ and $g$ . $\newline$ Line $f$ $\newline$ \begin{tabular}{|c|c|} $\newline$ \hline $x$ & $y$ \\ $\newline$ \hline $2$ & $7$ \\ $\newline$ \hline $4$ & $10$ . $5$ \\ $\newline$ \hline $7$ & $15$ . $75$ \\ $\newline$ \hline $11$ & $22$ . $75$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ Line $g$ $\newline$ \begin{tabular}{|r|r|} $\newline$ \hline $x$ & $y$ \\ $\newline$ \hline $-3$ & $4$ \\ $\newline$ \hline $-2$ & $0$ \\ $\newline$ \hline $1$ & $-12$ \\ $\newline$ \hline $4$ & $-24$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ Which system of equations represents lines $f$ and $g$ ? $\newline$ F $\newline$ $\begin{array}{l} y=1.75 x+3.5 \\ y=-4 x-8 \end{array}$ $\newline$ G $\newline$ $\begin{array}{l} y=1.75 x+3.5 \\ y=-4 x-2 \end{array}$ $\newline$ H $\newline$ $\begin{array}{l} y=3.5 x+1.75 \\ y=-4 x-8 \end{array}$ $\newline$ J $\newline$ $\begin{array}{l} y=3.5 x+1.75 \\ y=-4 x-2 \end{array}$

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Full solution

24

The tables of ordered pairs represent some points on the graphs of lines

f

and

g

\newline

Line

f

\newline

\begin{tabular}{|c|c|}

\newline

\hline

x

y

\newline

\hline

2

7

\newline

\hline

4

10

5

\newline

\hline

7

15

75

\newline

\hline

11

22

75

\newline

\hline

\newline

\end{tabular}

\newline

Line

g

\newline

\begin{tabular}{|r|r|}

\newline

\hline

x

y

\newline

\hline

-3

4

\newline

\hline

-2

0

\newline

\hline

1

-12

\newline

\hline

4

-24

\newline

\hline

\newline

\end{tabular}

\newline

Which system of equations represents lines

f

and

g

\newline

\newline

\begin{array}{l} y=1.75 x+3.5 \\ y=-4 x-8 \end{array}

\newline

\newline

\begin{array}{l} y=1.75 x+3.5 \\ y=-4 x-2 \end{array}

\newline

\newline

\begin{array}{l} y=3.5 x+1.75 \\ y=-4 x-8 \end{array}

\newline

\newline

\begin{array}{l} y=3.5 x+1.75 \\ y=-4 x-2 \end{array}

Calculate Slope for Line f: To find the equation of line f, we need to determine its slope $m$ and y-intercept $b$ from the given points. The slope can be calculated using any two points from the table. Let's use the points $(2, 7)$ and $(4, 10.5)$ . The slope $m$ is given by the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Calculating the slope: $m = \frac{10.5 - 7}{4 - 2} = \frac{3.5}{2} = 1.75$ .
Find Y-Intercept for Line f: Now, we need to find the y-intercept $b$ of line f. We can use the slope we found and one of the points to solve for $b$ using the equation $y = mx + b$ . Let's use the point $(2, 7)$ and the slope $m = 1.75$ . Substituting the values into the equation: $7 = 1.75 \times 2 + b$ . Solving for $b$ : $b = 7 - 3.5 = 3.5$ .
Equation for Line f: The equation for line f is $y = 1.75x + 3.5$ . Now, let's find the equation for line g using the same method. We'll calculate the slope using the points $(-3, 4)$ and $(-2, 0)$ . The slope $m$ is given by the formula $m = (y_2 - y_1) / (x_2 - x_1)$ . Calculating the slope: $m = (0 - 4) / (-2 - (-3)) = (-4) / (1) = -4$ .
Calculate Slope for Line g: Next, we need to find the y-intercept $b$ of line g. We can use the slope we found and one of the points to solve for $b$ using the equation $y = mx + b$ . Let's use the point $(-2, 0)$ and the slope $m = -4$ . Substituting the values into the equation: $0 = -4 \times (-2) + b$ . Solving for $b$ : $b = 0 - 8 = -8$ .
Find Y-Intercept for Line g: The equation for line g is $y = -4x - 8$ . Comparing the equations we found for lines $f$ and $g$ with the given choices, we see that the correct system of equations is: $y = 1.75x + 3.5$ and $y = -4x - 8$ . This corresponds to choice $F$ .