Choose the system of equations that matches the following graph
The Correct Answer and Explanation is:
The correct choice is the first option:
x – 4y = -4
3x + 4y = 36
Here is a detailed explanation of how to arrive at this answer.
To find the system of equations that matches the graph, we need to determine the individual equation for each of the two lines shown. We can do this by identifying key features of each line, such as its slope and y-intercept, directly from the graph.
Analysis of the Blue Line
First, let’s find the equation for the blue line, which has a positive slope (it rises from left to right).
- Identify two points: We can identify two clear points that the blue line passes through on the grid. For instance, it passes through the x-intercept at (-4, 0) and the y-intercept at (0, 1).
- Calculate the slope (m): The slope is the “rise over run,” or the change in y divided by the change in x. Using the points (-4, 0) and (0, 1):
m = (y₂ – y₁) / (x₂ – x₁) = (1 – 0) / (0 – (-4)) = 1 / 4.
So, the slope of the blue line is 1/4. - Determine the y-intercept (b): The y-intercept is the point where the line crosses the y-axis. From the graph, we can see the blue line crosses the y-axis at y = 1. So, b = 1.
- Write the equation: Using the slope-intercept form, y = mx + b, we can write the equation for the blue line:
y = (1/4)x + 1 - Convert to standard form: The answer choices are in the standard form Ax + By = C. To convert our equation, we can first multiply the entire equation by 4 to eliminate the fraction:
4y = x + 4
Next, we rearrange the terms to match the format of the options:
-x + 4y = 4
Multiplying the entire equation by -1 gives us:
x – 4y = -4
This equation matches the first equation in the correct answer choice.
Analysis of the Red Line
Now, let’s find the equation for the red (or dark) line, which has a negative slope (it falls from left to right).
- Identify two points: We can see that this line passes through its y-intercept at (0, 9) and another clear point at (4, 6).
- Calculate the slope (m): Using the points (0, 9) and (4, 6):
m = (y₂ – y₁) / (x₂ – x₁) = (6 – 9) / (4 – 0) = -3 / 4.
The slope of the red line is -3/4. - Determine the y-intercept (b): The line crosses the y-axis at y = 9, so b = 9.
- Write the equation: Using the slope-intercept form, y = mx + b:
y = (-3/4)x + 9 - Convert to standard form: Again, we convert to the form Ax + By = C. First, multiply by 4 to clear the fraction:
4y = -3x + 36
Now, add 3x to both sides to bring the x and y terms together:
3x + 4y = 36
This equation matches the second equation in the first answer choice.
Conclusion
The system of equations we derived from the graph is:
x – 4y = -4
3x + 4y = 36
This system perfectly matches the first option provided. As a final check, we can verify that the intersection point of the two lines, which appears to be (8, 3), satisfies both equations.
For x – 4y = -4:
(8) – 4(3) = 8 – 12 = -4. (Correct)
For 3x + 4y = 36:
3(8) + 4(3) = 24 + 12 = 36. (Correct)
Since the intersection point satisfies both equations, we have confidently identified the correct system.
