Which graph matches the linear equation: −2x+3y=−6
The linear equation −2x+3y=−6 can be rearranged to the form y=mx+b, where m is the slope and b is the y-intercept.
Rearranging the equation, we get:
3y = 2x - 6
y = (2/3)x - 2
The corresponding graph should have a slope of (2/3) and a y-intercept of -2.
Option C matches this criteria.
The tables below show the number of jumping jacks completed after a given period of time in seconds. Kimberly: Time (seconds) Jumping Jacks 3 17 8 37 12 53 16 69 Katrina: Time (seconds) Jumping Jacks 2 10 5 25 12 60 20 100(3 points) a. Which person is doing more jumping jacks per second? b. Which person had done more jumping jacks initially before the timer started? c. Which person shows a proportional relationship?
a. The person doing more jumping jacks per second can be determined by finding the average rate of change of jumping jacks over time.
For Kimberly:
The average rate of change between 3 seconds and 16 seconds is (69-17)/(16-3) = 4 jumping jacks per second.
For Katrina:
The average rate of change between 2 seconds and 20 seconds is (100-10)/(20-2) = 5 jumping jacks per second.
Therefore, Katrina is doing more jumping jacks per second.
b. To determine which person had done more jumping jacks initially before the timer started, we can look at the first point in each table.
For Kimberly:
Kimberly had done 17 jumping jacks at 3 seconds.
For Katrina:
Katrina had done 10 jumping jacks at 2 seconds.
Therefore, Kimberly had done more jumping jacks initially before the timer started.
c. A proportional relationship exists when the ratio of the number of jumping jacks to the time elapsed remains constant.
For Kimberly:
The ratio of jumping jacks to time elapsed ranges from 17/3 to 69/16. The ratios are not constant, so there is no proportional relationship.
For Katrina:
The ratio of jumping jacks to time elapsed ranges from 10/2 to 100/20. The ratios are constant (5/1), so there is a proportional relationship.
Therefore, only Katrina shows a proportional relationship.
To identify the graph that matches the linear equation −2x + 3y = −6, we need to rearrange the equation into the slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept.
Start by isolating the y variable:
−2x + 3y = −6
3y = 2x - 6
Divide both sides by 3:
y = (2/3)x - 2
Now we can compare the equation with the slope-intercept form to identify the slope and y-intercept:
The slope, m, is 2/3
The y-intercept, b, is -2
Now let's discuss the possible graphs that match this equation:
Graph 1: Line with a slope of 2/3 and a y-intercept of -2:
This line moves up two units for every three units moved to the right. It intersects the y-axis at -2.
Graph 2: Line with a slope of 2/3 but does not pass through the point (0, -2):
This line has the same slope but different y-intercept.
Based on the description, it seems that "Graph 1: Line with a slope of 2/3 and a y-intercept of -2" is the graph that matches the linear equation −2x + 3y = −6.
To find the graph that matches the linear equation −2x + 3y = −6, we need to rearrange the equation into slope-intercept form, which is in the form y = mx + b. In this form, 'm' represents the slope of the line, and 'b' represents the y-intercept.
Let's rearrange the equation −2x + 3y = −6 to get it in slope-intercept form:
Starting with −2x + 3y = −6:
Add 2x to both sides: 3y = 2x - 6
Divide all terms by 3: y = (2/3)x - 2
Now, y = (2/3)x - 2 is in slope-intercept form, where the slope 'm' is 2/3 and the y-intercept 'b' is -2.
To graph this linear equation, we start with the y-intercept, which is -2. This means the point (0, -2) is on the graph.
Next, we use the slope to determine the direction of the line. The slope of 2/3 means that for every 3 units moved horizontally (to the right), we need to go up 2 units vertically.
Starting from the y-intercept (0, -2), we can plot another point. Moving 3 units to the right, we land on (3, 0). Now, we can draw a line passing through both points.
Thus, the graph that matches the linear equation −2x + 3y = −6 is a straight line that passes through (0, -2) and (3, 0).