Choose the system of equations that matches the following graph
The Correct Answer and Explanation is:
The correct system of equations is:
x - 4y = -4
3x + 4y = 36This corresponds to the first option.
Explanation
To find the system of equations that matches the graph, we need to determine the equation for each of the two lines. We can do this by finding the slope and y-intercept of each line.
1. The Blue Line
First, let’s find the equation for the blue line which has a positive slope (it goes up from left to right).
- Find the y-intercept (b): The line crosses the y-axis at the point (0, 1). So, the y-intercept is b = 1.
- Find the slope (m): We can pick two points on the line to calculate the slope using the formula m = (y2 – y1) / (x2 – x1). Let’s use the y-intercept (0, 1) and another clear point, such as (4, 2).
- m = (2 – 1) / (4 – 0) = 1 / 4.
- Write the equation: Using the slope-intercept form y = mx + b, the equation is y = (1/4)x + 1.
- Convert to standard form (Ax + By = C):
- Multiply by 4 to remove the fraction: 4y = x + 4.
- Rearrange the terms: -x + 4y = 4, which is equivalent to x – 4y = -4.
2. The Red Line
Next, let’s find the equation for the red line which has a negative slope (it goes down from left to right).
- Find the y-intercept (b): The line crosses the y-axis at the point (0, 9). So, the y-intercept is b = 9.
- Find the slope (m): Let’s use the y-intercept (0, 9) and another point on the line, like (4, 6).
- m = (6 – 9) / (4 – 0) = -3 / 4.
- Write the equation: Using y = mx + b, the equation is y = (-3/4)x + 9.
- Convert to standard form (Ax + By = C):
- Multiply by 4: 4y = -3x + 36.
- Rearrange the terms: 3x + 4y = 36.
Conclusion
The system of equations derived from the graph is:
- x – 4y = -4
- 3x + 4y = 36
This matches the first option provided. We can also verify this by checking the point of intersection, which is (8, 3). Plugging x=8 and y=3 into the first option’s equations confirms it is the correct solution.
