Math  /  Algebra

Question\begin{align*} \text{Which system of equations best represents the table and the graph below?} \\ \begin{tabular}{cccccc} xx & -2 & -1 & 1 & 2 \\ yy & 0.5 & 1.75 & 4.25 & 5.5 \end{tabular} \\ \text{Equation Line } h \text{ (from the table):} \\ \text{Equation Line } j \text{ (from the graph):} \\ \text{The user took a picture with their phone and the text was extracted above. The user then had a dialogue with an AI Assistant to help clarify the instructions.} \\ \text{Dialogue Transcript:} \\ \text{Hi there! I see that we have a table with values and a graph referred to, but it seems like some crucial information is missing to fully solve the problem. Specifically, we need to know the equations for Line } h \text{ based on the table data and Line } j \text{ based on the graph.} \\ \text{Could you please provide either the missing equations or any additional information about the graph and how it relates to the problem? This will help me assist you better!} \\ \text{find slope of the graph} \\ \text{I would be happy to help you find the slope of the graph. However, since I can't see the graph, could you please provide a couple of points from the graph? This will allow me to calculate the slope for you. Once you provide the points, we can proceed with finding the slope.} \\ \text{(0, 5) and (4, 3)} \end{align*}

Studdy Solution

STEP 1

1. We have a table of values for Line h h .
2. We have two points for Line j j from the graph: (0, 5) and (4, 3).
3. We need to find the equations for both Line h h and Line j j .

STEP 2

1. Determine the equation for Line h h using the table values.
2. Determine the equation for Line j j using the given points from the graph.

STEP 3

Determine the equation for Line h h using the table values.
To find the equation of a line, we need the slope m m and the y-intercept b b . We can use the formula for the slope between two points:
m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}
Choose two points from the table, say (-2, 0.5) and (-1, 1.75):
m=1.750.51(2)=1.251=1.25 m = \frac{1.75 - 0.5}{-1 - (-2)} = \frac{1.25}{1} = 1.25
Now, use the slope-intercept form y=mx+b y = mx + b with one of the points, say (-2, 0.5), to find b b :
0.5=1.25(2)+b 0.5 = 1.25(-2) + b 0.5=2.5+b 0.5 = -2.5 + b b=3 b = 3
Thus, the equation for Line h h is:
y=1.25x+3 y = 1.25x + 3

STEP 4

Determine the equation for Line j j using the given points from the graph.
Use the points (0, 5) and (4, 3) to find the slope m m :
m=3540=24=0.5 m = \frac{3 - 5}{4 - 0} = \frac{-2}{4} = -0.5
Use the slope-intercept form y=mx+b y = mx + b with one of the points, say (0, 5), to find b b :
Since the point (0, 5) is on the y-axis, b=5 b = 5 .
Thus, the equation for Line j j is:
y=0.5x+5 y = -0.5x + 5
The system of equations that best represents the table and the graph is:
Line h h : y=1.25x+3 y = 1.25x + 3 Line j j : y=0.5x+5 y = -0.5x + 5

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